LIBRARY OF CONGRESS.* 
' ©1^. - ®op^ng|l %i 



UNITED STATES OF AMERICA. 



AN OUTLINE COT^RSE 



JVIECHA^IICAli) DRAWIJMG 



EVENING DRAWING CLASSES, 



Cflf- 



W. S. LOCKE, 

tl 

INSTRUCTOR IN MECHANICAL DRAWING, 



sf 



RHODE ISLAND SCHOOL OF DESIGN. 

;"^Pij 23 u 




PROVIDENCE, R. I. 

Copyright 1SS9 

and 1S91. 



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INDEX. 



INTRODUCTION. 
FREE-HAND DRAWING. 
PLANE GEOMETRY. 

Problems in Plane Geometry. 
SOLID GEOMETRY. 

Square Root. 

Cube Root. 
ORTHOGRAPHIC PROJECTIONS. 

Revolutions. 

True Length of a Line. 

Problems in Projections. 
CONIC SECTIONS. 

INTERSECTIONS and DEVELOPMENTS. 
ISOMETRIC PROJECTIONS. 
LINEAR PERSPECTIVE. 
WORKING DRAWINGS. 
DESIGN. 
CAMS. 

Face Cams. 

Square Cams. 

Edge Cams. 

Leaders. 
GEARING. 

Single Curve Gearing. 

Rack to Mesh with Single Ct^rve Gear. 

Single Curve Gears having less than 30 teeth. 



IV. INDEX. 

Bevel Gear Blanks. 
Epicycloidal Gears. 
Proportions of Gear Wheels. 
Teeth of Gear Wheels. 
Strength of a Tooth. 
Comparison of Teeth. 
Gear Designing. 

Sample Plate of Involute Teeth. 
TABLES AND DIAGRAMS. 
Strength of Materials. 
Factor of Safety. 
Horse Power of Shafting. 
Strength of Belting. 
Decimal Equivalents. 
Handle Table. 

Proportions of Bolts and Nuts. 
Alphabets. 
Areas of Circles. 
Circumferences of Circles. 



INTRODUCTION. 



This book is not a treatise on Mechanical Drawing, 
but an outline course on the subject. A mastery of its 
contents will not make the student an accomplished 
draughtsman, but it is hoped that such mastery will' 
help him a long way toward that desired result. 

The book is put out chiefly for that class of students^ 
who, from age or condition, are not able to take the 
time necessary to go through a complete course of 
study with the view of making themselves mechanical 
draughtsmen. The difficulty of gauging the needs of 
this body of students makes the success of any such 
effort problematical ; nevertheless, this trial is made 
with the hope of ultimate success. In attempting to 
meet the want referred to, the style of writing at- 
tempted, is forcible rather than elegant, direct rather 
than graceful. While much is lost by adopting this 
style, it is hoped that more is gained. 



*i 



FREE-HAND DRAWING. 



The student, like the child, must '' creep before he 
can walk,-' that is, do some preparatory work before 
he can stride along in his chosen path. In this pre- 
liminary work, the foundation, care should be taken. 
Study and work thoroughly, rather than fast. Make 
clear to your mind each subject as it is taken up, and 
by what is understood that which follows will be made 
easier. 

Before writing became common, the saying, *' The 
pen is mightier than the sword,'' was accepted by the 
world. In these days, when time is very valuable, 
drawings are generally used in all kinds of business. 
Drawing is the universal language, and the ability to 
make good drawings is of more value, to-day, than any 
mere physical power. More or less elaborate drawings 
are used to illustrate form, fancy, or thought, wherever 
man communicates with his fellow. The art of making 
pictures has grown from rude outlines made on bark of 
trees with charcoal to magnificent oil paintings in splen- 
did colors. It is not the object of this book to teach 
this art, but it will be well for the student of Mechanical 
Drawing to learn to make outline drawings, first, with- 
out instruments, other than pencil and paper. The 
ability to make these free-hand drawings is useful and 
valuable, always cultivating the eye and hand for more 
complicated work. Representations of simple forms 



MECHANICAL DRAWING. 



should be attempted at first, as even these may prove 
difficult to the beginner. IVe must learn to know what 
our eyes see and then to represent that correctly. 



Fio.l. 



R(?,^ . 




Figure i represents 
three '' views," top, 
side, and end of a 
simple block. The 
student should begin 
with some such sim- 
ple form, and learn 
to see and represent 
it correctly. To do 
this, place the block on 
the level of the eye, in 
such a position that only 
four lines, bounding one 
plane of the block, can 
be seen. These simple 
drawings having been made, the block should be placed 
lower and an attempt made to draw a " perspective" of 
it. Fig. 2 is an accurate perspective of the block, three 
of whose faces are shown in Fig. i . While the repre- 
sentation is correct, quickly recognized and simple, there 
are some details that will repay study. First, it is stand- 
ing on a level surface because certain lines are vertical. 
Second, the lines that form the bottom and top do not, 
in themselves, appear as level lines, while we know they 
must be. Third, the angles made at the meeting of the 
lines are not right angles — the corners do not appear as 
being square, though we know they are so. If the stu- 
dent sees and understands these three points, and also 
the fact that the drawing — Fig. 2 — is correct, he has 
taken a good step in the right direction. It will not be 
arrived at in an hour, and many times a day the attention 



MECHANICAL DRAWING. 9 

should be fixed on a form of definite outlines and an 
attempt made to see and represent it. This training- 
should continue until the student can draw, with fair 
speed and accuracy, such simple inanimate objects as he 
handles in his business. It will often happen that the 
ability to do this will save time and money to employer 
or employed, or both. The draughtsman is lame with- 
out it. The fundamental principles of perspective being 
thus understood, the student may continue to practice 
free-hand drawing both to his pleasure and profit, but 
for the purpose of accomplishing the work laid out in 
this Course, we may leave free-hand drawing for the 
next step. 

This may well be ' Geometrical Drawing, for the 
science which treats of Position and Extension must 
help us to correct such inaccuracies as we have now 
noticed in work where the eye is the only judge of 
correctness. 

INSTRUMENTS. 

Before we take up Geometrical Drawing we should 
know something of instruments and their use. The 
skill of hand already acquired will be of great use in 
handling instruments, and the accuracy of eyesight^ 
necessary in making measurements. Accurate 77ieasiu'e- 
ments are absolutely necessary to good Geometrical and 
Mechanical Drawing. A complete outfit for Mechanical 
Drawing may be bought for $15.00, and half this sum 
will buy instruments sufficient for most plain work. 

The following is a satisfactory outfit : One drawing 
board, one T-square, two triangles, one scale, two rub- 
bers, pencils, and case of instruments. The following- 
are good sizes: Board, twenty-four by thirty-two 
inches, T-square with thirty-inch blade, a six-inch rub- 
ber 45° triangle, a ten-inch rubber 30-and-6o° triangle, 



lO MECHANICAL DRAWING. 

a twelve-inch steel scale, pencil and ink eraser, one each, 
four thumb tacks, four and six H. Faber's drawing pen- 
cils, one case instruments, containing one five-and-one- 
half-inch compasses, with pen, pencil and needle points, 
one five-inch spring dividers, two right line pens, three- 
inch spring dividers, three-inch spring bow pencil and 
three-inch spring bow pen. This may come well 
within $15.00, and be composed of goods that will 
wear for many years. 

In using the T-square it should be placed on the board 
with its head held against the left hand side. All hori- 
zontal lines should be drawn with the pencil held against 
the upper side of the blade of the square as it is thus 
held. Perpendicular lines should be drawn with the 
combined help of T-square and triangle. With the 45*^ 
and the 30-and-6o° triangles the following angles, with 
a horizontal, may be obtained : 15°, 30°, 45°, 60^, 75 
and 90^. The student will profit by a facility in placing 
his triangles to make these angles. In the Geometry 
which follows, a method is given of dividing an angle in 
halves, so that the student should be able to make any 
desired angle. The most convenient scale is a steel 
scale, twelve inches long, graduated to sixteenths on one 
edge and thirty-seconds on the other. It is made by 
Brown & Sharpe, and sold everywhere for $1.25. 

The compasses are for making circles and arcs of cir- 
cles. In drawing a circle never take hold of a point of 
the dividers, but always hold the compasses by the head 
or handle. The joints. in the stocks are used to bend the 
points to nearly right angles with the paper when draw- 
ing large circles. In using the right-line pen always 
hold it in a plane at right angles to the paper. Always 
draw, never push the pen. This last is a good rule also 
in using the pencil, which should be sharpened to a 
chisel edge rather than to a point. This sort of '' point" 



MECHANICAL DRAWING. II 

is also best for the pencil used in the compasses. It will 
wear longer and make a better line. Much care should 
be taken in all these operations, for these are details of 
the art of drawing, which is composed of details. Ten 
times more time is lost erasing incorrect lines than is 
lost in careful decision as to whether or not they will be 
right. In drawing, it is very clearly seen that "haste 
makes waste." The first requisite of a drawing, is 
ACCURACY, the second, neatness. After these, in a 
sheet of drawings, comes, third, arrangement. 

With careful practice one may learn to draw fast, but 
no one can make good, reliable drawings in a hurry. 



PLANE GEOMETRY. 



DEFINITIONS. 

Geojnetry is the Science of Position and Extension, 

A Point has merely position j without extension. 

Extension has three dimensions : Le7igth^ Breadth and 
Thickness. 

A Line has only one dimension, namely, length. All 
lines are either straight or curved. 

Note. — In this book the word "line" is understood to mean a 
straight liyie^ unless otherwise specified. 

A Surface has two dimensions : Length and breadth. 

A Solid has the three dimensions of extension : length, 
breadth and thickness. 

The Position of a Point is determined by its Direction 
and Distance from any known point. 

A Plane is a surface in which any two points being 
taken, the straight line joining those points lies wholly 
in that surface. 

Axiom. A straight line is the shortest distance be- 
tween two points. 

THE ANGLE. 

An Angle is formed by two lines meeting or crossing 
each other. 



MECHANICAL DRAWING. I3 

The V'ertex of the angle is the point where its sides 
meet. 

The magnitude of the angle depends solely upon the 
difference of direction of its sides at the vertex or amount 
they are spread apart. 

The- magnitude of the angle does not depend upon 
the length of its sides. 

When one straight line meets or crosses another, so 
as to make the two adjacent angles equal, each of these 
angles is called a Right angle, and the lines are said to 
be perpendicular to each other. 

Thus the angles ABC and ABD (Fig. 3) being equal, 
are right angles. 

An Acute angle is one less than a right angle, as K 

(Fig. 3)- 

An Obtuse angle is one greater than a right angle as 

-V(Fig. 3)._ 

Parallel Lines are straight lines which have the same 
Direction, as AB^ CD (Fig. 3). 

Parallel lines cannot meet, however far they are pro- 
duced. 

POLYGONS. 

A plane figure is a plane terminated on all sides by 
lines. 

If the lines are straight, the space which they contain 
is called a rectilineal figure, ox polygon {F Fig. 3). 

The polygon of three sides is the simplest of these 
figures, and is called a tiHangle; that of four sides is 
called a quadrilateral ; that of five sides, ?i pentagon; that 
of six, a hexagon^ &c. 

A triangle is denominated equilateral {E Fig. 3), 
when the three sides are equal, isosceles (/ Fig. 3) , when 
two only of its sides are equal, and scalene (^9 Fig. 3), 
when no two of its sides are equal. 



H 



MECHANICAL DRAWING. 



A right-triangle is one which has a right angle. The 
side opposite to the right angle is called the hypothcmise. 
Thus ABC (Fig. 3) is a triangle right-angled at A, and 
the side BC is the hypothenuse. 

Among quadrilateral figures, w^e distinguish : 

The square (Sq. Fig. 3), having its sides equal, and 
its angles right angles. 

The rectangle {^R Fig. 3), having its angles right 
angles, and its opposite sides equal. 

T\\Q parallelogram (/* Fig. 3), which has its opposite 
sides parallel. 

The rhombus or lozenge {Rh. Fig. 3), which has its 
sides equal, w^ithout having its angles right angles. 



Rq.S. 




The trapezoid ( 7' Fig. 3), which has two only ot its 
sides parallel. 

A diagonal is a line which joins the vertices oi two 
angles not adjacent, as AC in the figure of the paral- 
leloijrani. 



MECHANICAL DRAWING. 15 



THE CIRCLE. 



Definitions, The circumfereitce of a circle is a curved 
line^ all the points of which are equally distant from a 
point within, called the centre. 

The circle is the space enclosed by this curved line. 

The radius of a circle is any straight line, as AB^ AC, 
AD (Fig. 3), drawn from the centre to the circumfer- 
ence. 

The diameter of a circle is a straight line, as \BD, 
drawn through the centre, and terminated each w^ay 
by the circumference. 

A semicircufiiference is one half of the circumference, 
and a semicircle is one half of the circle itself. 

An arc of a circle is any portion of its circumference, 
as BFE, 

The circle is supposed to be divided into 360 equal 
parts called degrees, marked ^ . Thus " an angle of 
90*^" is, of course, one-quarter of a circle. 60^, is one- 
sixth, 45^, one-eighth, &c. 

The chord of an arc is the straight line, as BE, which 
joins its extremities. 

The segment of a circle, is a part of a circle compre- 
hended between an arc and its chord, as EFB, 

A tangent (Fig. 3) is a line, which has only one point 
in common with the circumference, as HD. 

A polygon is said to be circumscribed about a circle, 
when all its sides are tangents to the circumference ; 
(Fig. 3,) and, in this case, the circle is said to be in- 
scribed in the polygon. 

A polygon is inscribed in a circle when all its vertices 
are in the circumference of the circle, (Fig. 3.) 



Problems in Plane Geometry. 



Note. — Lay out a sheet of drawing paper with tifteen equal 
rectangles inside the three-quarter inch margin. Write a problem 
and make a graphical solution in each rectangle In each problem 
there are certain things given and certain things required. Carefully 
note and make use of the things given in finding the things required. 

I. Problem. To find the position of a point in a 
plane, having given its distances from two known points 
in that plane. 

Solution. Let the known points be A and B. From 
the point ^ as a centre, with a radius equal to the dis- 
tance of the required point from ^, describe an arc. 
Also, from the point ^ as a centre, with a radius equal 
to the distance of the required point from B, describe 
an arc cutting the former arc ; and the point of intersec- 
tion C is the required point. 

a. By the same process, another point D may also 
be found which is at the given distances from A and B, 
and either of these points therefore satisfies the condi- 
tions of the problem. 

b. If both the radii were taken of equal magnitudes, 
the points C and D thus found would be at equal dis- 
tances from A and B, 

c. The problem Is impossible, when the distance 
between the known points is greater than the sum of 
the given distances or less than their diflerence. 



MECHANICAL DRAWING. I 7 

d. If the required point is to be at equal distances 
from the known points, its distance from either of them 
must be greater than half the distance between the 
known points. 

2. Problem. To divide a given straight line AB into 
two equal parts ; that is, to bisect it. 

Solution. Find by § b^ a point C, above the line, at 
equal distances from the extremities A and B. Find 
also another point D.^ below the line, at equal distances 
from A and B. Through C and D draw the line CD^ 
which bisects AB at the point E. 

3. Pi'oblem. At a given point A^ in the line BC^ to^ 
erect a perpendicular to this line. 

Solution. Take the points B and C at equal distances 
from A ; and find a point D equally distant from B and 
C. Join AD and it is the perpendicular required. 

4. Problem. From a given point A., above the 
straight line BC, to let fall a perpendicular upon this 
line. 

Solution, From A rs a centre, with a radius suffi- 
ciently great, describe an arc cutting the line BC in two 
points B and C ; find a point B> below BC\ equally dis- 
tant from B and C, and the line AB> is the perpendicular 
required. 

5. Problem. To draw a perpendicular to a line at 
one end. 

Solution. Let AB be a horizontal line* With A as 
centre and radius AB draw an arc. With same radius 
and centre B draw arc, cutting first one at D. With 
D as centre and same radius draw arc over A. Draw 
line through B and D^ meeting the last arc at /s. ^ Line 
EA will be perpendicular to line AB. 



iS MECHANICAL DRAWING. 

6. Problem. To make an arc equal to a given arc 
AB^ the centre of which is at the given point C. 

Solution. Draw the chord AB. From any point D 
as a centre, with a radius equal to the given radius CA^ 
describe the indefinite arc FH. From 7^ as a centre^ 
with a radius equal to the chord AB, describe an arc 
cutting the arc FH \\\ H, and we have the arc FH^=l 
arc AB. 

7. Problem. At a given point A., in the line AB^ to 
make an angle equal to a given angle K. 

Solution. From the vertex A", as a centre, with any 
radius, describe an arc IL meeting the sides of the angle ; 
and from the point ^ as a centre, by the preceding prob- 
lem, make an arc BC equal to IL. Draw AC, and we 
have angle BAC = angle K. 

8. Problem. To bisect a given arc AB. 

Solution. Find a point D at equal distances from A 
and B. Through the point F> and the centre C draw the 
line CP>^ which bisects the arc AB at F. 

9. Problem. To bisect a given angle A. 

Solution. From ^ as a centre, with any radius, de- 
scribe an arc BC^ and by the preceding problem, draw 
the line AF to bisect the arc BC^ and it also bisects the 
angle A. 

10. Problem. Through a given point A, above a 
given straight line BFC^ to draw a straight line parallel 
to the line BFC. 

Solution. Join FA, and, by Problem 7, draw AD^ 
making the angle FAD — AFC, and AD is parallel to 
BFC 



MECHANICAL DRAWING. I9 

1 1 . Problem. Two sides of a triangle and their in- 
cluded angle being given, to construct the triangle. 

Solution. Make the angle A equal to the given angle ^ 
take AB and AC equal to the given sides, join BC^ and 
ABC is the triangle required. 

12. Problem, The three sides of a triangle being 
given, to construct the triangle. 

Solution. Draw AB equal to one of the given sides^ 
and, by § I, find the point C at the given distances AC 
and BC from the point C, join AC and BC, and ABC is 
the triangle required. 

Note. — The problem is impossible, when one of the given sides is. 
greater than the sum of the other two. 

13. Problem. The adjacent sides of a parallelogram 
and their included angle being given, to construct the 
parallelogram . 

Sohttion. Make the angle A equal to the given angle ^ 
take AB and AC equal to the given sides, find the point 
Z^, by § I, at a distance from B equal to AC, and at a 
distance from C equal to AB. Join BD and DC^ and 
ABCD is the parallelogram required. 

Note. — If the given angle is a right angle, the figure is a rectangle; 
and, if the adjacent sides are also equal, the figure is a square. 

14. Problem. To find the centre of a given circle or 
of a given arc. 

Solution. Take at pleasure three points. A, B, C, on 
the given circumference or arc ; join the chords AB and 
BC^ and bisect them by the perpendiculars DE and FG ; 
the point O in which these perpendiculars meet is the 
centre required. 



20 MECHANICAL DRAWING. 

15. Problem. Find by the same construction as in 
Problem 14 a circle, the circumference of which passes 
through three given points not in the same straight line. 

16. Problem. Through a given point to draw a 
tangent to a given circle. 

Solution. If the given point A is in the circumference, 
draw the radius CA^ and through A draw ^Z) perpendic- 
ular to CA^ and AD is the tangent required. 

17. Problejn. Through a given point to draw tan- 
gents to a given circle. 

- Solution. If the given point A is without the circle, 
join it to the centre by the line AC ; upon AC as a diam- 
eter describe a circumference cutting the given circum- 
ference in J/ and A/' ; join AM and AA^^ and they are 
the tangents required. 

18. Problem. To inscribe a circle in a given triangle 
ABC. 

Solution. Bisect the angles A and B by the lines AO 
and BO, and their point of intersection O is the centre 
of the required circle, and a perpendicular let fall from 
O upon either side is its radius. 

Note. — The three lines AO, 7?0 and TO, which bisect the three 
angles of a triangle, meet at the same point. 

19. Problem. To divide a given straight line AB 
into any number of equal parts. 

Solution. Suppose the number of parts, for example, 
is six. Draw the line AO, making an acute angle with 
line AB\ take AC of any convenient length and apply 
it six times to AO. Join B and the last point of divis- 
ion, Z>, by the line BD. Through the last point of 
division but one draw a line (see Problem 10) parallel 
to BD. This line will cut AB, at E, one-sixth of the 
distance from B toward A. Apply EB six times to AB, 



MECHANICAL DRAWING. 21 

20. Problem. To find a mean proportional between 
two given lines. 

Solution. Draw the straight line ACB, making AC 
equal to one of the given lines, and EC equal to the 
other. Upon ACB as a diameter describe the semicircle 
ADB. At C erect the perpendicular CZ^, and CD is the 
required mean proportional. 

21. Problem. To divide a given straight line ACB 
at the point C in extreme and 7nean ratio, that is, so that 
we may have the proportion : 

AB : AC=AC: CB. 

Solution. At end B erect the perpendicular BP> equal 
to half of ACB. Join AP>, take DP from D on AD 
equal to BD, and AC equal to AP, and C is the required 
point of division. 

REGULAR POLYGONS. 

Defi7titions. A regular polygon is one which is at the 
same time equiangular and equilateral. 

Hence the equilateral triangle is the regular polygon 
of three sides, and the square the one of four. 

An equilateral polygon is one which has all its sides 
equal ; an equiangular polygon is one which has all its 
angles equal. 

22. Problem. To inscribe a square in a given circle ► 

Solution. Draw two diameters, AB and CD, perpen- 
dicular to each other ; join AD, DB, BC, CA ; and 
ADBC is the required square. 

23. Problem. To inscribe a regular hexagon in a 
given circle. 

Solution. Take the side BC of the hexagon equal to 
the radius ^C of the circle, and, by applying it six 



22 MECHANICAL DRAWING. 

times round the circumference, the required hexagon 
BCDEFG is obtained. 

24. Pj'oblem. To describe a regular decagon in any 
circle. 

Sohctioii. Divide the radius of a (three-inch) circle 
in extreme and mean ratio. (Problem 21.) 

The longer part is equal to one side of the regular 
decagon required. Apply it ten times to the circum- 
ference, and join the points by straight lines, making the 
decagon. 

Aiake a pentagon by joining the alternate vertices of 
the decagon. 

25. Problem. To circumscribe a circle about a given 
regular polygon ABCD^ &c. 

Solution. Find, by Problem 14, the circumference of 
a circle w^hich passes through three vertices, A^ B, C ; 
and this circle is circumscribed about the given polygon. 

26. Pi^oblem. To inscribe a circle in a given regular 
polygon A BCD, &c. 

Solution. Bisect two sides of the polygon by perpen- 
diculars, the point of intersection is the centre of the 
required circle. 

The sides of the polygon become tangents to the 
circle. 

27. Problem. To inscribe a pentagon in a circle. 

Solution. Draw a diameter AB and a radius CD per- 
pendicular to it. Bisect BC at E. With centre at E 
and radius ED draw arc DF^ — F being on AC. With 
Z> as a centre, and DF as a radius, draw arc EG, meet- 
ing the circumference at G. Draw line DG. It is one 
side of the required pentagon. 

28. Problem. To construct a regular polygon of any 
number of sides in a circle (Approximate method). 



MECHANICAL DRAWING. 23 

Solution. Draw a diameter AB and divide it into as 
many equal parts as there are sides in the required poly- 
gon—say eight. With A and B as centres, and radius 
AB draw arcs intersecting in C. Draw line from C 
through the second point of division of AB to meet the 
circumference at D. AD is one side of the required 
polygon. 

AREAS. 

Definitions. Equivalent figures are those which have 
the same surface. 

The area of a figure is the measure of its surface. 

The unit of surface is the square w^hose side is a 
linear unit ; such as a square inch or a square foot. 

The area of a square is the square of one of its sides. 

A parallelogram is equivalent to a rectangle of the 
same base and altitude. 

The area of a parallelogram is the product of its base 
by its altitude. 

Parallelograms of the same base are to each other as 
their altitudes ; and those of the same altitude are to 
each other as their bases. 

All triangles of the same base and altitude are equiva- 
lent. 

The area of a triangle is half the product of its base 
by its altitude. 

Every triangle is half of a parallelogram of the same 
base and altitude. 

The circumference of a circle is equal to 3. 141 6 multi- 
plied by its diameter., or r. D. 

The area of a circle is equal to 3. 14 16 multiplied by 
the square of its radius., or r: R\ 

The area of a trapezoid is half the product of its alti- 
tude by the sum of its parallel sides. 



24 MECHANICAL DRAWING. 

29. Problem. To construct a square equal to three 
times a given square. 

Solution. Extend one side of the given square, and 
lay off on it the length of its diagonal. Draw a line 
from the point at which this diagonal ends to the extreme 
angle of the square, and upon this line construct a square, 
w^hich will be the square required. 

30. Problem. To make a square equivalent to the 
sum of two given squares. 

Solution. Construct a right angle C ; take CA equal 
to a side of one of the given squares ; take CB equal to 
a side of the other ; join AB, and AB is a side of the 
square sought. 

A square may be found equivalent to a given triangle, 
by taking for its side a mean proportional between the 
base and half the altitude of the triangle. 

A square may be found equivalent to a given circle, 
by taking for its side a mean proportional between the 
radius and half the circumference of the circle. 

PRACTICAL PROBLEMS IN PLANE 
GEOMETRY. 

Note. — Divide the sheet into four equal parts and put a problem 
in each. 



Make an angle of 75° and bisect it. 



2. Draw six-ineh parallel lines, y<^//r inehes apart. 

3. Divide a seven-ineh line into nine equal parts. 

'4. Make a simple belt pulley withy^z'^ spokes. 

5. Make an isosceles triangle equivalent to a right- 
angled triangle whose sides are three., four ?iwdi five inches 
long. 



MECHANICAL DRAWING. 25 

6. Make an isosceles triangle equivalent to a par- 
allelogram whose sides are three^ and foicr inches^ and 
whose angles are 6d^ and 120°. 

7. Make a square equivalent to a right-angled trian- 
gle whose sides are three, four and five inches, 

8. Make a square equivalent to a circle whose diame- 
ter \% four inches. 



SOLID GEOMETRY. 



DEFINITIONS. 

From the definitions of Plane Geometry we may 
recall at this time those of a Line', a Surface, a Plane, 
and a Solid. 

A Solid has the three dimensions of extension ; length, 
breadth and thickness. 

Every solid bounded by planes is called ?<. polyedron. 

The bounding planes are called the faces ; whereas 
the sides or edges are the lines of intersection of the 
faces. 

A polyedron of four faces is a tetraedroji^ one of six 
is a hexaedrou^ one of eight is an octaedron^ one of twelve 
a dodecaedron^ one of twenty an icosaedrou, &c. 

The tetraedron is the most simple of polyedrons ; for 
it requires at least three planes to form a solid angle, 
and these three planes leave an opening, which is to be 
closed by a fourth plane. 

A prism is a solid comprehended under several par- 
allelograms, terminated by two equal and parallel poly- 
gons. 

The bases of the prism are the equal and parallel 
polygons. 

The convex surface of the prism is the sum of its 
parallelograms. 



MECHANICAL DRAWING. 



27 



The altitude of a prison is the perpendicular 
distance between its bases. 

A right prism is one whose lateral faces or 
parallelograms are perpendicular to the bases. 



J^.. 



Fig. 3A. 



Note. — In this book the word prism is to be taken to mean a 
right prism ^ that is one whose lateral faces are rectangles. 

In this case each of the sides has an altitude equal to that of the 
prism. 

A prism is triangtdar ^ quadrangular.^ pentagonal^ hex- 
agonal., &c., according as its base is a triangle, a quad- 
rilateral, a pentagon, a hexagon, &c. 

The prism, whose bases are regular polygons of an 
infinite number of sides, that is, circles, is called a 
cylinder. 

The line which joins the centres of its bases is called 
the axis of the cylinder. 

In the right cylinder the axis is perpendicular 
to the bases, and equal to the altitude. 

The right cylinder may be considered as gen- 
erated by the revolution of a rectangle about one 
of its sides. The other side generates the con- 
vex surface, and the ends generate the bases of 
the cylinder. 

A prism whose base is parallelogram has all its faces 
parallelograms, and is called a parallelopiped. 

When all the faces of a parallelopiped are rectangles, 
it is called a right parallelopiped. 

^ The cube is a right parallelopiped, com- 
prehended under six equal squares. 

The cube, each of whose faces is the unit 
of surface, is assumed as the ////// of solidity. 

Fig. 30. 




Fig. 3B. 



28 MECHANICAL DRAWING. 

The volume^ solidity^ or solid cojitents of a solid, is the 
measure of its bulk, or is its ratio to the unit of solidity. 

The cubic inch is a good unit of solidity. 

The area of the convex surface of a prism or cylinder 
is the perimeter^ or the circumference of its base inulti- 
plied by its altitude. 

The formula for the convex surface of a cylinder is,. 
6' = t: DH^ where S^^ the convex surface, 7rz=3.i4i6,. 
D = diameter, and If = height. 

Careful attention is called to this mode of expression,, 
for it is convenient and will be used hereafter. 

In the expression S = t: DH, the quantities written 
together are supposed to be multiplied together, thus 
S= t: Z>Zr becomes S= tt x jDxIf. 

Any prism or cylinder is equivalent to a right prisn> 
of the same base and altitude. 

The solidity or volume of any piHsm or cylinder is the 
product of its base by its altitude. 

Prisms or cylinders of equivalent bases and equal 
altitudes are equivalent. 

Let R be the radius, and A area of the base of a cylin^ 
der; and r. — 3.1416, we have A = 3.i4i6x^'. 

Denoting, also, the altitude by H and the solidity or 
volume of the cylinder by V^ we have V=AxJf = -X 
R'xII=r.R'H. 

A pyramid is a solid formed by several triangular 
planes proceeding from the same point and terminating 
in the sides of a polygon. This point is the vertex or 
apex of the pyramid. 

The altitude of the pyramid is the distance of its vertex 
from its base. 

A pyramid is regular when the base is a 
regular polygon, and the perpendicular let 
fall from the vertex upon the base, passes, 
through the centre of the base. This per- 

PiG^ pendicular line is the axis of the pyramid. 





MECHANICAL DRAWING. 29 

When the base of a pyramid is a circle it 
is called a <:^;^<f. The <^.t/j" of the cone is the 
line drawn from the vertex to the centre of the 
base. 

Pig. L. 

A right cone is one, the axis of which is perpendicular 
to the base. 

The right cone may be considered as generated by the 
revolution of a right triangle about one leg as an axis ; 
the other leg generates the base, and the hypothenuse 
generates the convex surface. 

The area of the convex sm^face of a right cone is half 
the product of the circumference of the base, by the 
side. We may reduce this rule to a formula, if we let 
the surface be represented by S^ diameter of the base by 
Z^ and the side by H\ as follows : -S = >^ -DH\ 

As before, it must be remembered that the quantities 
written together are multiplied together. 

The volume or solidity of a cone is one- third of the 
product of its base by its altitude or height. Let H 
represent the height. The area of the base is 3.1416X 
R^ ; hence the volume of a cone is F = 1/371 R^H. 

The volume of any pyramid is one -third of the product 
of its base by its height. 

Pyramids or cones of equivalent bases and equal alti- 
tudes are equivalent. 

Any pyramid or cone is a third part of a prism or 
cylinder of the same base and altitude. 

THE SPHERE. 

DEFINITIONS. 

A Sphere is a solid terminated by a curved surface, all 
the points of which are equally distant from a point 
within called the centre. 

*3 




30 MECHANICAL DRAWING. 

The sphere may be conceived to be gen- 
erated by the revolution of a semicircle about 
its diameter. 

The radius of a sphere is a straight line 
Fig. E. draw^n from the centre to a point in the sur« 
face ; the diameter or axis is a line passing through the 
centre, and terminated each way by the surface. 

All the radii of a sphere are equal ; and all its diam- 
eters are also equal, and double the radius. 

Every section of a sphere made by a plane is a circle. 

The section made by a plane w^hich passes through the 
centre of the sphere is called a great circle. Any other 
section is called a small circle. 

The radius of a great circle is the same as that of the 
sphere, and therefore all the great circles of a sphere are 
equal to each other. 

The centre of a small circle and that of the sphere are 
in the same straight line perpendicular to the plane of 
the small circle. 

The area of the surface of a sphere is the product of 
its diameter by the circumference of a great circle. 

The surface of a sphere is equivalent to four great 
circles. 

The surfaces of spheres are to each other as the 
squares of their radii. 

From the foregoing we may deduce the following 
formulas : 

Surface of a sphere :=^ S ^ 4' R^ ^^ - Z>^, for since 
the area of one circle is -i?^ the surface of a sphere is 
4 X -K'^4-R\ 

The solidity or volume of a sphere is one-third of the 
product of its surface by its radius. 

The surface — S= 4- R" multiplied by Yi R\'$> V — 
{4' R') X Yz R = 7: ^T.R\ or Volume of sphere = V = 



MECHANICi\L DRAWING. 3 1 

For the convenience of the student, rules for extract- 
ing the Square and Cube Roots are here inserted. 

SQUARE AND CUBE ROOT. 

RULE. 

To extract the square root of any number : 

1 . Beginning with the units figure, point off the ex- 
pression into periods of two figures each. 

2. Find the greatest square in the number expressed 
by the left hand period, and write its square root as the 
first figure of the 7^oot. 

3. Subtract this square from the part of the number 
used, and to the remainder unite the next two terms of 
the given number for a new dividend. 

4. Double the part of the root already found for a 
trial divisor ; and by it divide the new dividend — less the 
last figure — and write the quotient as the next figure of 
the root. Also, write it at the right of the trial divisor^ 
the combined figures making the true divisor. 

5. Multiply the true divisor by the last figure of the 
root and subtract the product from the new dividend. 

6. If there are any more terms of the root to be 
found, unite with the remainder the next two terms of 
the given number, and take for a trial divisor, double the 
root already found, and proceed as before. 

RULE. 

To extract the cube root of a number : 

I. Beginning with the units figure, point off the ex- 
pression into periods of three figures each. 



32 MECHANICAL DRAWING. 

2. Find the greatest cube in the number expressed 
by the left hand period, and write its cube root as the 
first figure of the root. 

3. Subtract the cube from the part of the number 
used, and with the remainder unite the next three figures 
of the given number for a new dividend. 

4. Take three times the square of the part of the root 
already found, with two ciphers annexed, for a trial 
divisor, and by this divide the new dividend, and write 
the quotient as the next term of the root. 

5. To the trial divisor add three times the first term 
of the root.^ with a cipher annexed, multiplied by the 
last term, also the square of the last term of the root, 

6. Multiply this sum by the last term of the root^ and 
subtract the product from the new dividend. 

7. If there are more terms of the root to be found, 
unite with the remainder the next three figures of the 
given number, take for a trial divisor three times the 
square of the part of the root now found, and proceed as 
before. 



ORTHOGRAPHIC PROJECTIONS. 



All Mechanical Drawing is founded on Mathematics — 
principally on Plane, Solid and Descriptive Geometry. 

We have now some knowledge of Plane and Solid 
Geometry. 

Descriptive Geometry is that branch of Mathematics 
which has for its object the explanation of the methods 
of representing, by drawings, all geometrical magni- 
tudes ; also, the solution of problems relating to these 
magnitudes in space. 

Drawings are so made as to present to the eye, situ- 
ated at a particular point, the same appearance as the 
magnitude or object itself, were it placed in the proper 
position. The representations thus made are the p7'o- 
jections of the object. 

The planes upon which these projections are usually 
made are the planes of projectio7i. The point at which 
the eye is situated is the point of sight. 

Definition. When the point of sight is in a perpen- 
dicular, drawn to the plane of projection, through any 
point of the drawing, and at an infinite distance from 
this plane, the projections are Orthographic. 

This result is reached, physically, if we suppose the 
eye to be as large as the object and placed in the perpen- 
dicular referred to, and at any convenient distance. 

Definition. When the point of sight is within a finite 
distance of the drawing, the projections are Sccnographic, 
commonly called the Perspective. 



34 



MECHANICAL DRAWING. 




The student should gain a sound knowledge of Ortho- 
graphic projection before attempting Perspective, hence 
our attention will be directed to the former for the 
present. 

In Orthographic project- 
ions, three planes of pro- 
jections ( sometimes two 
suffice) are used, at right 
angles to each other, one 
horizontal and the other 
two vertical, called respect- 
ively the horizontal and ver- 
tical planes of projection, 
and denoted by the letters H and V. 

Let a rectangular cross (Fig. 5) ,be imagined self- 
suspended near a lower corner of a room, or between 
three sheets of paper, placed in a similar position, name- 
ly, at right angles to each other ; the three principal 
dimensions, length, breadth and thickness, of the cross 
being each perpendicular to one of the sheets of paper— 
which serve as the three planes of projection. As indi- 
cated by the dotted lines, let perpendiculars be drawn 
from the principal points of the cross to each plane of 
projection. 

Let the two vertical sheets be now laid down on a 
table, keeping the top of the cross in line on both. Now 
(Fig. 4) we have the three projections of the cross on 
one plane in the manner in which it is proper to repre- 
sent them as Orthographic projections. 

It is easily seen that neither of these projections is a 
correct representation of the cross as we see it, and, also, 
that collectively the three projections truly represent the 
cross in length, breadth and thickness. Here, then, is 
the value of this method of representing objects. 



MECHANICAL DRAWING. 



35 



All that these projections need to make them working 
drawings^ are the dimensions in figures. The projection 
on H is called the Plan^ and the two on V and F, are 
called Elevations. 









^ 




























V 


!• 1 \ 




\- 


\^ 














/ 











The representation (in Fig. 5) is in Perspective. Fig- 
ures I and 2 represent the whole principle in the same 
manner. 

GROUND LINE. 

It is to be observed that the line which (in Fig. 5) is 
made by the intersection of H and F, is preserved in 
Fig. 4. It is called the Ground Line. The representa- 
tion of the object above the Ground Line is called an 

« 

Elevation, and the one below is called the Plan, of the 
object. 

Returning now to Fig. 5, it will be seen that the Plan 
is drawn upon the plane you look down upon, and the 
elevations upon planes you look upon horizontally. 

After a little experience, the Ground Line becomes as 



36 



MECHANICAL DRAWING. 



imaginary as the Equator, but like the latter serves its 
purpose. 

^. V. g. 9 IG, il 




Figure 6 represents in Plan and Elevation a triangular 
prism ; Fig. 7, a rectangular prism ; Fig. 8, a square 
pyramid ; Fig. 9, a hexagonal pyramid ; Fig. 10, a right 
cylinder, and Fig. 11, a cone. 

These names, objects and representations should be 
kept in mind, for they will be referred to many times. 

This subject of Orthographic projections is the most 
important of all subjects to the mechanical draughts- 
man. He uses it a thousand times to one of any other 
method of representation, and should be proportionally 
well acquainted with it. In all that follows in this book 
this acquaintance will be cultivated, until, it is hoped, 
any line in space may be comprehended and drawn. 

The subjects which immediately follow, are especially 
cho.sen to develop facility in making Orthographic pro- 
jections, as well as to gain accurate knowledge of various 
solids and their combinations. 

REVOLUTIONS. 

So far we have confined ourselves to projections of 
objects placed at right angles to the planes of projection, 



MECHANICAL DRAWING. 37 

but it will be easily understood that in making drawings 
of machines or houses, we shall find many lines which 
are not so related to natural planes of projection already 
described. For an example, let us take the rectangular 
prism (Fig. 7) just used. To assist us in the right wa}^, 
we figure the corners of the front side (Fig. 12). In the 
plan the figures double, but make no confusion so long 
as we have the elevation to look at. 

Tip the prism, now, so that the base line 3, 4 will 
make an angle of 30"" with the ground line, kee^oing the 
plane of the face i, 2, 3, 4, parallel to the ground line. 

To make a second plan of the prism in this second 
position : As the different representations or views of 
an object are supposed always to be in positions perpen- 
dicular to each other, the corner i, for example, will be 
found in a perpendicular to ground line. As the prism 
was not inclined to the vertical plane, the desired corner 
will be found in a line through i of the plan, parallel to 
the ground line. 

The perpendicular from i in the second elevation, and 
the parallel from i in the first plan meet, making i of 
the second plan. 

In the same way seven other corners are found and 
the new plan finished. 

We have now a plan and elevation of the prism as it 
is inclined to the GL. Move this second plan to the 
right and incline it to the GL at an angle of 45 ^\ Now 
from the third plan draw a perpendicular, and from the 
second elevation draw a parallel, from corner i. The 
point in which these two lines meet is corner i in tlie 
third elevation. One by one seven other j^oints may be 
found, completing the elevation of the prism as it ap- 
pears inclined to both planes of projection. 

This work brings us to a point where we may attempt 
the problem : " To find the true length of a line." 




n 



38 MECHANICAL DRAWING. 

TRUE LENGTH OF A LINE. 

RULE. 

Revolve one of the projections of the line until it is 
parallel to the Ground Line. Construct the other pro- 
jection of the line to agree with the second position. 
This construction will be the true length of the line. 

EXPLANATION. 

Let ;//, ;/, represent the horizontal, 
and ;// n the vertical, projections of 
-a line. To proceed according to the 
rule : Revolve ///, ;/, about ;;/ until // 
is at ;/" and ;//, ;/' is parallel to G L. 
In this revolution it is supposed that 
the angle of the line with the hori- p^Q p 

zontal plane is not changed. If this be so, ;/' has not 
changed its height above the horizontal plane, and its 
new position is to be found at the intersection of a 
parallel from n and a perpendicular from ;/", or at N. 
.;;/' not having moved, the true length of the liur must 
be ;;/' N. 

For practice, take the two projections of the line i, 4, 
in the third plan and elevation of the prism just drawn 
in " revolutions." Revolve the elevation of the line and 
construct the plan. The accuracy of the work may be 
tested by the first elevation of the line. 

PROBLEMS. 

1. Find the true length of a line whose elevation ap- 
pears as a line 4" long, inclined 45^ to the Ground Line, 
and whose plan has an angle of 30^ to the G. L. 

2. Draw two views of a rectangular prism 3 x 2 x t 
and give the true length of its diagonal. 



MECHANICAL DRAWING. 39 

3. Draw two projections of a hexagonal pyramid, 
having a base inscribed in a circle whose diameter is 
three inches, and whose altitude is five inches. Give 
the true length of the centre line of one side. 

4. Draw a hexagonal pyramid exactly like that of 
Problem 3, and give the true length of the line which 
joins the middle point of the middle line of a side, to 
the centre of the base. 



CONIC SECTIONS. 



The Conic Sections are so called because they are sec- 
tions of a cone. 

We have had a definition of a plane. Imagine two 
such surfaces passed through a solid, at a distance from 
each other of less than the thousandth part of an inch. 
The slice of the solid between the planes is termed a 
Section, It is also called a lamina^ or a slice section. 
Also, we often use the term Section when but one plane 
is passed. 

The Conic Sections are taken from a right cone and 
are, the Triangle, Circle, Ellipse, Parabola and Hyper- 
bola. The Triangle is a section cut from a cone by two 
planes passed through the apex cutting the base. 

A Circle is a section of a right cone cut at right angles 
to the axis. 

The Ellipse is a curved section cut at any angle to the 
axis, large enough to cut both sides of the cone. 

The Hyperbola is a curved section cut from the cone 
parallel to the axis and perpendicular to the base. 

The Parabola is a curved section cut from a right cone 
parallel to one of the sides, as it appears in elevation. 

Of these Sections, the triangle and circle are in con- 
stant use, in drawing ; the ellipse in frequent use ; the 
parabola occasionally, and the hyperbola but rarely. 



MECHANICAL DRAWING. 



41 




Fig. G. 



To learn, practically, what these curves are and how 
to get them from a cone, proceed as follows ; 

Make a plan and elevation of a 
cone, having a base four and a half 
inches in diameter and six and one- 
half inches high* In the elevation 
draw the elevations of the Sections, 
as represented. The plans of but 
two of the Sections are given, for 
more would make confusion. It will 
be easily understood that F G \^ the 
base of the hyperbola, and D E the 
base of the triangle. These are in 
their true lengths. The altitudes of 
all the Sections are given in the ele- 
vation of the cone ; that of the hyper- 
bola, for example, is H L. If, now, 
we erect a perpendicular, equal to H Z, to the middle of 
a base line equal to F G, the points corresponding to F, 
L and G will be three important points in the hyperbola. 
If, now, we want more points, upon which to construct 
the curve, we proceed as follows : 

Pass a plane through the cone parallel to the base, 
cutting the hyperbola. It will cut a circle from the 
cone, as N^ O. This circle is on the convex surface of 
the cone. As the hyperbola is also upon its surface they 
must intersect at A. Drawing, now, the plan of circle 
JV O from centre X, we find that the width of the circle, 
at A^ is B C. But the two curves intersect at A. Hence 
B C IS also the width of the hyperbola at a distance, 
jET A^ from the base. Set off H A on the axis of the 
hyperbola, from the base, and draw a line parallel to the 
base. On this line set off ^ S^ each side, frona the axis. 
These points will be points in the hyperbola. More 



42 



MECHANICAL DRAWING. 



planes must be passed, more circles drawn, and more 
points obtained to be accurate, especially near the top of 
the curve. The parabola and ellipse are obtained in 
the same w^ay. In the parabola care must be taken to 
set off R F on the axis and the perpendicular from F, 
that is, F Z as the width, of the parabola at height R F^ 

X E F> IS, the plan of the triangle. It is to be drawn 
" full size," but no directions are given, as the student is 
expected to work out for himself the major part of these 
Sections. A knowledge gained by zuork is retained and 
used, when frequently careful instruction is forgotten. 
As the ellipse occurs frequently in drawing and requires 
expensive instruments for its delineation, the following 
approximate method, by arcs of circles, is given : 

Draw major axis A 
B^ and half minor axis 
CD. Complete the rect- 
angle A B X a. Draw 
A D. Draw a b, per- 
pendicular to C D. Ex- 
tend C n to e b. With 
radius C D draw arc 
D /. With diameter 
A f draw semicircle A 
e f. Draw radius g h. T 

Lay off h /on c b to / F^^- H. 

Through j draw an arc with b as centre. From A as 
centre and radius C e draw an arc, cutting arc through/ 
at k. Through k draw bl, and through m draw ko : in 
is a centre for A o, k centres for o /, and b, for ID. A o 
I D \^ one-quarter of the ellipse. 

The parabola may be described as follows : 
Suppose the parabola to have a base CD and an alti- 
tude A B. Extend A B to E, making B E equal to BA. 




MECHANICAL DRAWING. 



43 



Draw E C and E D. Divide C E and E D into any 
number of equal parts, numbering 
one from the top and the other 
from the bottom. Join i, i?^ — 
2, 2, — 3, 3, &c. These lines will 
all be tangents to the parabola. 
With a French curve the para- 
bola may be drawn. In drawing 
an irregular curve with a French 
curve in this way, be sure that 
the instrument touches three of 
the points through which the 
curve is to be drawn. 




Intersections and Developments. 



As the memory will easily recall, there are many lines 
seen on a manufactured article, or on a drawing of it, 
that are not lines of any individual part of the article, 
but lines that occur where two or more forms intersect 
or join each other. Such lines are called intersection 
lines. 



H 




Intersections is the name given 
to that part of geometrical 
drawing that treats of the in- 
tersection lines and their cor- 
rect delineation. 

It v^ill be seen at once that a 
thorough knowledge of geo- 
-metrical solids will be neces- 
sary to the student who desires 
to take a full course in inter- 
sections. On the other hand, 
we all have a fair understanding 
of many geometrical forms that meet our eyes in com- 
bination every day, and with these we will make a 
beginning. 

Let A B^ C D^ represent two projections of a hexagonal 
prism, and -fi", 3, 6, ZT, i, 4, two projections of a hex- 
agonal pyramid, passing through, or intersecting the 
prism. To learn what the intersection line will be, 
proceed as follows : 




MECHANICAL DRAWING. 



45 



First, number the angles of the base of the pyramid so 
that the position of a line in both projections maybe easily 
noted. The line i-E intersects the prism at Z, whose 
elevation is at K, on line i H. The line 6-E intersects 
the prism at J/, whose elevation is at JV j hence the in- 
tersection of the face Zr,-i,-6, of the pyramid, with the 
prism, is the line K JV. ^-E is directly under 6-E^ so 
its elevation will be, at O. By similar reasoning, the 
intersecting point on /\-H is at P, and K-N-O-P is the 
intersection li7te. It will easily be seen that the similar 
line on the other side of the prism is constructed in the 
same way. It will be good practice for the student to 
copy the plan of the combination, with the line \E at an 
angle of 15^ with the Ground Line, and then construct 
the elevation. 

We will suppose a triangular prism passed through a 
right cylinder (Fig. 13). 

First make a plan and two elevations of the cylinder* 
In the middle of the elevation which is projected from 
the plan, draw the end elevation of the prism. 




Now draw the plan, assuming that the prism projects 
from the cylinder at either side, and that centre line of 
prism and cylinder coincide ii;i Z>, E^ C. 



46 MECHANICAL DRAWING. 

In the side elevation it is evident that there will be an 
intersection line ; that none appears in the plan and end 
elevation is evident because in tlie plan it coincides w^ith 
the outline of the cylinder, and in the end elevation it 
coincides with the outline of the prism. 

We can lay out the top and bottom lines and ends, 
on the side elevation, from the other views. The point 
where tlie lower line of the prism pierces the cylinder is 
found as follow^s : 

In the plan draw the line Z, 7V^, perpendicular to the 
diameter Z, O^ through the point J/, where the prism 
pierces the cylinder. 

Lay off the distance Z, O from X to F, the latter being 
the desired point. 

The top point of the intersection line is on the cir- 
cumference. To find other points in the intersection 
line, pass the planes i and 2 and proceed in the same 
w^ay as in the case of the bottom line. 

It is now required to develop the surface of the cylin- 
der. In Mechanical Drawing this means to draw an 
equivalent plane figure. This may be illustrated by 
fitting the surface of the cylinder with a covering of 
paper. When this paper is unrolled and spread on a 
table, we have a surface equivalent to that of the cylin- 
der. It is now required to outline, in this develop7nent 
of the cylinder, the hole that the prism makes. Let A^ 
H (Fig. 13) represent part of the development of the 
cylinder. Let a perpendicular at Z>, represent the axis 
of the prism. Lay ofi' the arc Z, J/, developed as a 
straight line on each side of the axis, making the line 
R^ P. The third corner is found on the axis at the alti- 
tude of the prism from R^ P. Intermediate points are 
laid oft^ from the axis on traces of the planes i and 2 in 
the same manner. 



MECHANICAL DRAWING. 



47 



We have had the intersection of plane surfaces with 
plane surfaces, plane surfaces with curved surfaces, and 
now we have curved with curved surfaces. 




Fig. 13 A illustrates the intersection of two cylinders. 
The method of finding the intersection line is as fol- 
lows : In the plan pass the plane i' through the cylin- 
ders parallel to both axes. It will cut a rectangle from 
each. ^/ is one end of the rectangle cut from the 
smaller cylinder, and c f o- // is the elevation of the 
rectangle in its true size. To find the rectangle cut 
from the larger cylinder, draw a semicircle ABC, show- 
ing one-half of the ii\\(\ of the c\ Under. The trace of 



48 



MECHANICAL DRAWING. 



the plane across it is seen in the line 3, 4, which, of 
course, is one-lialf the end of the rectangle. This half 
laid off each way from D^ in the elevation, at a a^ shows 
VIS that the rectangle \% a a a a. We see that these rect- 
angles intersect at X and three other points, which must 
be intersection points, that is, points which are in both 
solids, for they were cut by one plane, y and other 
points are found by the same process. To find point Z, 
pass a plane through A tangent to smaller cylinder. 
The rectangle in the smaller cylinder is reduced to a 
single line, and the rectangle of the larger cylinder in- 
tersects it at Z. The curved intersection line G xyz may 
now be drawn by free-hand, by a template whittled from 
thin wood, or by a French curve. Duplicated three 
times, it will complete the elevation as seen in the plate. 

It is now required to 
develop the cylindrical sur- 
face of the larger cylinder. 
The semicircle ABC^ Fig. 
13 A, is the basis of our 
work. Set a pair of spac- 
ing dividers closely and 
step round the curve. 
Suppose that it is 20 
steps. In Fig. 13 B these 
20 steps will make the 
straight line T B^ which 
double.d gives the full 
development B B. TIV 
is, of course, the length 
of the cylinder. To find 
tlie hole made in this 
cylinder, lay off H G as 
T G^ ^7 A as 7^7? ^nd by 
as 7^8. Planes i, 2, in 




MECHANICAL DRAWING. 



49 



Fig. 13 B, are then in same position as in Fig. 13 A. 
With the dividers space o^ HJV and transfer it to V S at 
9, H O 2X 10, and H P ?it z. Planes i' 2' through 9 and 
10 will then have the same position on the surface as in 
Fig. 13 A. The points x and jf v^ill be at the intersec- 
tion of the traces of the planes, and G and z on the cen- 
tre lines, and Gxyz will be the developed curve, and 
may be duplicated to complete the figure. It will not 
be a true ellipse, departing from that curve as shown 
at Q. 

In all these intersections there is but one method 
used — that of passing a plane through the solids as 
they appear in one view and then constructing, in 
another view, the plane surfaces cut out of the solids, 
from vs^hence the intersections of the planes give points 
in the intersections of the solids. Once have a clear 
idea of what a plane is, and how it should be passed, 
and the subject of intersections becomes easy. 

A hint of the method of passing the plane : Always 
cut the solids by the planes so as to get the simplest pos- 
sible figures. 

In making developments the ability to get the true 
length of any line, shown by two or more projections, 
will save time and assist toward accuracy. 

In making working drawings, or other drawings of 
combined solids, these intersection lines constantly ap- 
pear, and facility in their representation is a necessity to 
the draughtsman. The workman also is greatly bene- 
fited by a knowledge of them, which, in short, are parts 
of the forms which he handles every day. 



ISOMETRIC PROJECTIONS. 



Prof. Farish, of Cambridge, England, in 1820, gave 
the term Isometrical Perspective to a particular projec- 
tion, which represents a cube from a position where 
J:hiee sides appear as equal rhombuses. 

The term Isometric means equal measure. 

Let three straight lines be drawn, intersecting in a 
common point and perpendicular to each other, two of 
them being horizontal aud the third vertical — like the 
three adjacent edges of a cube. 

Then let a fourth straight line be drawn through the 
same point, making equal angles with the first three, as 
the diagonal of a cube. If, now, a plane be passed per- 
pendicular to this fourth line, and the straight lines and 
other objects be orthographically projected upon it, the 
projections are called Isonetric. 

The three straight lines first drawn are the co-ordinate 
axes ; and the planes of these, taken two and two, are 
the co-ordinate planes. The common point is the origin. 
The fourth line is the Iso7netric Axis. 

Since the co-ordinate axes make equal angles with 
each other, and with the plane of projection, it is evi- 
dent that their projections will make equal angles with 
•each other, two and two, that is, angles of i3o°. Hence, 



MECHANICAL DRAWING. 5r 

(Fig. 14,) if any three straight lines, as Ax, Ay and Az^ 
be drawn through the point A, making with each other 
angles of 120°, these may be taken as the projections 
of the co-ordinate axes, and are the directrices of the 
drawing. 

It is further evident, that if any equal distances be 
taken on the co-ordinate axes, or on lines parallel to 
either of them, their projections will be equal to each 
other, since each projection will be equal to the distance 
itself into the cosine of the angle of inclination of the 
axes with the plane of projection. 

The angle which the diagonal of a cube makes with^ 
either adjacent edge is known to be 54^ 44'; therefore, 
the angle which either edge, or either of the co-ordinate 
axes, makes with the plane of projection will be the 
complement of this angle, viz., 35° 16'. 

If a scale of equal parts be constructed, the unit of the 
scale being the projection of any definite part of either 
co-ordinate axis, as one inch, or one foot, will be one 
inch multiplied by the natural cosine of 35^ 16'. We 
may from this scale determine the true length of the 
isometric projection of any given portion of either of" 
the co-ordinate axes, or of lines parallel to them, by 
taking from the scale the same number of units as the 
number of inches or feet in the given distance. Con- 
versely, the true length of any line in space may be 
found by applying its projection to the Isometric scale, 
and taking the same number of inches or feet, as the 
number of parts covered on the scale. 

Or: The isometrical length of a line, is the true 
length of a line multiplied by the natural cosine of 3^^ 
i6\ Now this cosine is .816; hence, if we multiply 
the true length of a line — say one inch long— by .Sr6, 
we will get the isometrical length of the line, that is, 



52 



MECHANICAL DRAWING. 



one inch multiplied by .8x6, which equals .8x6 (thou- 
sandths) of an inch, or about (yf) thirteen sixteenths of 
an inch. 

Now, if we have a 35^ 16' triangle, and a scale thir- 
teen-sixteenths full size, we have the special tools neces- 
sary to make an isometrical drawing. 

A much easier way is generally adopted. 

It is customary to use a 30^ triangle in place of a 35° 
16' triangle, and a full size scale in place of a |f scale. 

Since in most of the frame work connected with ma- 
chinery, and in various kinds of buildings, the principal 
lines to be represented occupy a position similar to the 
co-ordinate axes, namely, perpendicular to each other^ 
one system being vertical, and two others horizontal, the 
Isometric projection is used to great advantage in their 
representation. A still greater advantage arises from 
the fact that in a drawing thus made, all lines parallel to 
the directrices are constructed on a full size scale. 

If the isometrical projection of a point be required^ 
the following operation is sufficient : 




Thus in Fig. 14, let A Z, A V and A X he the direc- 
trices, A being the projection of the origin. On A^ F, 
lay o?i A, F equal to the distance of the point from the 
co-ordinate plane X, Z. 

Through P draw P, M' parallel to A, Z, and make it 
equal to the distance of the point from the plane X Y. 



MECHANICAL DRAWING. 53 

Through M^ draw M\ M parallel to AX, and make 
it equal to the third given distance, and M will be the 
required projection. 

The projection of any straight line parallel to either of 
the co-ordinate axes may be constructed by finding, as 
above, the projection of one of its points, and drawing 
through this, a line parallel to. the proper directrix. 

If the line is parallel to neither of the axes, the pro- 
jections of its ends may be found, as above, and joined 
by a straight line, which will be the projection required. 

The projection of curves may be constructed by find- 
ing a sufficient number of the projections of their points. 

PROBLEM. 

To construct the Isometric projection of a cube : Let 
the origin be taken at one of the upper corners of the 
cube, the base being horizontal, and let AX^ A Y and 
AZ (Fig. 14), being the directrices. 

From A, on the directrices, lay ofl^ the distances AX\ 
AZ and A F, each equal to the length of the edge of the 
cube. 

These lines will be the projections of the three edges 
of the cube which intersect at A, 

Through X, Y and Z draw Xe, Xg, Ye, Yc, Zc and Zg, 
parallel to the directrices, completing the three equal 
rhombuses A, X, e,Y, etc. 

These will be the projections of the three faces of the 
cube— which are seen — and the representation will be 
complete. 

It must be very carefully noted that only lines of 30^ 
with the horizon, and perpendicular lines, may be meas- 
ured on an isometrical drawing. All other lines arc more 
or less distorted. The following is a good method for 
drawing 

*5 



54 



MECHANICAL DRAWING. 



AN ISOMETRIC CIRCLE. 




Fig. K. 



Draw ABCD with 30^ triangle. Bisect AD at /. 
Erect perpendicular HX. With radius HO describe arc 
OX. With radius OX describe arc EX. OE is one- 
half longer axis. Bisect OE at G, Through G draw 
JN. With radius NJ, draw arc JK. Trisect OE at 
/andi^. With radius FE describe arc EL. Lay off 
FE on JG from / to M. With radius JH, centre AI, 
describe arc cutting EL at L. Through F draw LFFy 
with centre F describe arc LJ, completing one-quarter 
of the ellipse. Transfer centres and complete. 

Isometrical projection is especially valuable to the 
architectural draughtsman, as it explains many con- 
structions that could hardly be done by plans, elevations 
and sections, and it also unites with pictorial represen- 
tation, the applicability of a scale. For drawings for 
the Patent Office it is a convenient and easy method, 
combining the requisites of many projections ; but as a 
drawing of what could absolutely be seen by the eye, it 
is not truthful, and, therefore, when pictorial illustration 
o?ily is requisite, the drawing should be made in linear 
perspective. 



LINEAR PERSPECTIVE. 



From the definition of Orthographic projections which 
^ve have had, we understand that they can never present 
to the eye of an observer a perfectly natural appearance, 
and hence this mode of representation is used only in 
drawings made for the purposes of mechanical or archi- 
tectural constructions. Whenever an accurate picture 
of an object is desired, the scenographic method must be 
used, and the point of sight chosen where it would 
naturally be placed in looking at the object repre- 
sented. That application of the principles of Descrip- 
tive Geometry which has for its object the accurate 
representation, upon a single plane, of the details of the 
form and the principal lines of a body, is called Linear 
Perspective. The art by which a proper coloring is 
given to all parts of the representation, is called Aerial 
Perspective. This, properly, forms no part of a Course 
like this, and so is left entirely to the taste and skill of 
the artist. 

The surface upon which the representation of a body 
is made, is called the plane of the picture j the plane of 
the picture is usually taken between the object to be 
represented and \\\q. point of sight, in order that the draw- 
ing may be of smaller dimensions than the object. It is 
also taken vertical, as in this position it will, generalh', 
be parallel to many of the important lines of the object. 
The Orthographic projection of the point of sight on 



56 MECHANICAL DRAWING.* 

the plane of the picture, is called the principal point of 
tJic picture ; and a horizontal line through this point and 
in the plane of the picture, is the horizon of the picture. 

Observation has made it evident that the greatest angle 
under wdiich one or more objects can be distinctly seen 
is one of 90*^. If between the object and the eye there 
be interposed a transparent plane, (such as one of glass) 
the intersections of the visual rays (the visual rays are 
those reflected rays of light from the object to the eye 
which make it visible,) with this plane are termed per- 
spectives of the points from which the rays come. 

Referring now to Plate A, we use two horizontal 
lines in drawing the Perspective of an object. One is 
the intersection of the picture plane with the ground, 
and is called the Ground Line {G. L.) ; the other is the 
Horizon Line^ (already mentioned) and is located par- 
allel with, and five feet above the G. L. This relation 
of these two lines never changes as to direction, but 
always changes to conform with the scale of any par- 
ticular picture. It is supposed to be at the same height 
from the ground as the eye of the observer, and the 
Orthographic projection of the point of sight becomes 
the principal point of the picture {F, P.), The point 
of sight, or Point of the Observer {P. O.)^ is selected at 
will, in a perpendicular to the plane of the picture at 
the P. P. The point P. O. being selected, its distance 
from P. P. is laid oft' on the Horizon Li fie each side of 
P P. These points on the horizon line are termed van- 
ishing points of diagonals or Distance Points. If lines 
be drawn from these points to P. O., the included angle 
will be found to measure 90^, 

All lines drawn through the object, at an angle of 45^ 
with the picture plane, vanish at the distance points. All 
perpendiculars to the picture plane, through the object, 
vanish at the principal point. 



MECHANICAL DRAWING. 



57 



PROBLEM. 

To construct the perspective of a Square. Place the 
Orthographic plan of the square H E A F as far in front 
of' the G. L. as it is supposed to be behind it. First 
make a perspective of the diagonal F E. To do this, 
continue F E to O. The perspective of this diagonal 
is — as we have already learned — O E>. To get the per- 
spective of the perpendicular AE, extend it to B. Its 
perspective is B P. Consequently, the perspective of E^ 
which is at the intersection oi F O and A B, is at X, the 
intersection of the perspectives oi F O and AB. From 
the plate it is seen that the perspectives of the other cor- 
ners of the square are obtained in the same way. Hence, 
the 

RULE. 

The perspective of a point is found at the intersection of 
the perspectives of a diagonal and a perpetidicular through 
the point. It should be noticed that the perspective of 
the diagonal F E \?> the diagonal X Z. 




58 



MECHANICAL DRAWING. 



The application of this simple rule enables us to con- 
struct the perspective of any line lying on the ground, 
as, for example, the circle A B C D E F G H \^^ in per- 
spective, the ellipse i 2345678. The process of con- 
structing the perspective circle is simply that outlined in 
the rule, as will be easily understood from the plate. 

In making perspectives of solids, we use the rule 
given with Plate A. To find. the perspective of a point 
not on the ground, proceed as follows : 

The points of intersection with the G. L. of a diagonal 
and a perpendicular through the point, are transferred to 
a horizontal line located r:S far above the G. L. as the 
point is above the ground. From these last points, the 
perspective of the point is constructed as it would be 
from the G. L. 

In Plate B, the perspective of a point B is found seven 




MECHANICAL DRAWING. 



59 



feet from the ground, by transferring the points x^ y, on 
the G. L. to line O F, seven feet from the ground, and 
proceeding as in constructing point 2. The representa- 
tion is of a square prism seven feet high. This prism 
and the square in Plate A are represented in parallel 
perspective — one side being parallel to the G L. 




6o MECHANICAL DRAWING. 

The second representation on Plate B — that of a prism 
two feet square and three feet long, is in angular per- 
spective. The method used is that given in the descrip- 
tion of Plate A. The prism is placed in an awkward 
position purposely, to show that the rule given is ample 
for all conditions. 

In Plate C, we find a new kind of line, namely, one 
inclined to both the picture plane and the ground. Its 
perspective is found by our one rule. It is necessary to 
make two elevations first, from which the heights of 
points are taken. Architects use this method. By 
using two distance points and placing the plan at an 
angle of 45° with the G. Z., much work is saved, e. g., 
w^hen A i;s located, half the work of finding other eave 
corners is done. Note the method of finding centres of 
v^all surfaces by diagonals. 

Plate C shows the proper method of procedure in con- 
structing the perspective of any object. By it, the great 
object of properly placing the thing to be represented, 
is gained, and the representation appears natural, though 
much more room is required to perform the operations. 



WORKING DRAWINGS. 



It is the opinion of the author that a fair understand- 
ing of the principles touched upon in previous chapters 
is necessary to enable an ordinary draughtsman to do his 
work ; and, conversely, having such understanding, it will 
be easy for him to learn to make good working drawings. 
A working drawing is a drawing made for workmen to 
use in making the thing drawn. A working drawing 
should represent clearly the form, material and dime?isions 
of the thing to be constructed. The design must be 
made perfectly correct, because from it are taken the 
dimensions for the working drawing. Dimensions 
should be taken several times before being put on a 
drawing, and the workman required to work from the 
dimensions. This requirement is necessary, because 
the paper on which a drawing is made will shrink and 
swell with the changes in the weather, making it un- 
trustworthy as a guide to correct figures. It will be 
noted from what has been said, that the most important 
part of a working drawing is the dimensions. 

A sheet of working drawings should always bear a 
title, if possible > in the lower right hand corner. The 
title should answer with utmost brevity the following 
questions: What is it? What scale? When and by 
whom ? Some manufacturers require, in addition, the 
name of the firm and the number of sheets of working 
drawings necessary to represent all parts of the machine. 

6 



62 



MECHANICAL DRAWING. 




Figure 15 is a working 
drawing for a small jour- 
nal or box. A few points 
may be made from it. 
The figures are placed 
where they would nat- 
^1"^ urally be looked for.— 
They are made much 
larger and clearer than 
figures would be on an 
ordinary engraving. A 
workman desires to see 
these figures at a glance, 
not to be obliged to hunt for them. They are placed in 
a way to obscure the outline of the box the least possi- 
ble. The lines drawn for the index points are usually 
in red and do not obscure the real drawing as much as 
the black ones in this drawing do. 

This matter of putting dimensions upon working 
drawings seems to be the chief stumbling block of 
inexperienced draughtsmen. A knowledge of how 
the thing is made is the very greatest assistance in 
making a working drawing. In running out the red 
dimension lines, care should be taken that they run 
perpendicularly to the surface or line whose dimension 
is to be given. Index points and figures are always in 
black ink. Small dimensions are generally clearer if 
indicated as in the dimension ^" in Fig. 15. 

The distances between finished surfaces are important 
and should be put on first. There are various ways of 
indicating a surface that is to be finished, one of the 
simplest being writing the italic /across the line. Dis- 
tances between or from centre lines are generally impor- 
tant though often forgotten. The matter of centre lines 
itself is fundamental in making a working drawing. 



MECHANICAL DRAWING. 



63 



The centre lines of each view should be drawn first, and 
the remainder of the drawing laid out from them. A 
tapped hole should be marked T, as well as the thread 
indicated. In making three views of a curved line, plot 
the third from the other two rather than guess at it, for 
incorrect drawings are pretty sure of being expensive — 
in reputation to the draughtsman, and in money to the 
manufacturer. In the detail drawing of the Head of a 
Screw Slotting Machine which accompanies this subject 
some of these points may be noted. 

-3H> 




o\i \ 



K^ 



HEAD 

SCREW SLOTTER 

l^z^lFt. /2-2G-90 

WSL 



S 



iU 



S^ 



Fig. 16. 



The plan is turned improperly that the curves may be 
in full lines. It should, according to rule, be inverted- 
Points are taken on the curves in the two elevations and 
are transferred to the plan, where, by chance, they fall 
nearly into an arc of a circle. The student's drawing 



64 MECHANICAL DRAWING. 

from which this illustration was taken had an o-gee 
curve in place of this quarter circle. The centres of 
the arcs, forming the curves of the sides of the head, 
should always be given. Almost any curve may be 
matched up with arcs of circles, and irregular or French 
curves should not be used for outlines, for the curves 
are difficult of duplication by the pattern maker, and so 
lose time and money, through the draughtsman, where 
it should be saved. On the contrary, where radii are 
given, the pattern maker may easily get his outlines, 
whatever the scale of the drawing. 

It will be noticed that some of the lines of the draw- 
ing are heavier than others. These heavier lines are 
supposed to cast shadows from the direction of 45^ over 
the left shoulder of the draughtsman. 

In the plan this light " goes off" from the solid above 
and to the right of it, and on the elevation, at the right 
and bottom. In details the plan is generally shaded as 
are the other views because the objects may be placed in 
one position as well as another. It is not well, how- 
ever, to always follow the rigid rule of shading right 
hand and bottom lines. For example, a cylinder, if 
standing on end, should have the bottom line shaded, 
but if it were lying down, the bottom line should not be 
shaded, for the darkest line on a cylinder is some dis- 
tance from the edge. 

Much more sound instruction might be written on the 
art of making working drawings, but this is only an 
"Outline Course" in drawing, and this subject is dis- 
missed with this parting warning : Do not think of the 
state of the drawing after it has been in the shop a week, 
but make it as neatly, clearly and distinctly as you can. 



MECHANICAL DRAWING. 



65 




»6 



66 



MECHANICAL DRAWING. 




MECHANICAL DRAWING. 



67 




DESIGN 



Having learned to make working drawings, there is 
a natural desire to take the next step, and learn to design 
the buildings or machines that require the explanatory 
working drawing. The designing of cams and gearing 
will be explained, and a finished design is shown of a 
machine designed to roll hot bar steel into finished forg- 
ings for the market. While it is not the custom to shade 
such designs, the present one is so finished to give the 
student some instruction in that art. 

In the subjects just mentioned much will be learned, 
incidentally, of designs, in a class of facts that soon be- 
come common-place to the draughtsman or designer, 
but are as necessary to him as his paper. 

A good working table of strength of materials is 
given, also diagrams for ascertaining strength of shaft- 
ing and belting, together with data useful in connection 
with these tables and diagrams ; but it is beyond the 
scope of this work to teach a subject that requires origi- 
nal thought and oftentimes original research. 

Prof. R. H. Thurston says : * " The work of design- 
ing metal parts of machinery involves the intelligent 
consideration of the cheapest and most satisfactory 
methods of moulding those which are to be cast, as 

* Materials of Engineering, Part II., Article 144. 



MECHANICAL DRAWING. 69 

well as of forging parts made of wrought iron and steeL 
The pattern maker must also know how to prepare the 
pattern so as to avoid the difficulties frequently met with 
in moulding. The moulder is required to know how to 
mould the piece in order to secure sound castings ; and 
the founder must understand the mixing and melting of 
metals in such a manner as will give castings of the re- 
quired quality. The engineer should know what forms 
can be cheaply made in cast metal, and what cannot be 
cast without difficulty, or without liability to come from 
the mould unsound. He should be able to instruct the 
pattern maker in regard to the form to be given the pat- 
tern in order to make the moulder's work easy and satis- 
factory, to tell the moulder how to mould the pattern, 
with what to ffil his flask, how to introduce the molten 
metal, and to provide for the escape of air, gas, and 
vapor, and he should be able to specify to the founder 
the brands and mixtures of iron to be chosen." 

There is one suggestion to be made to the beginner 
in designing which is of supreme importance to him, 
namely : that the moving parts ^ not the frame, is the 
machine— in the same sense that the spirit of a man, 
not his body, is the man. It is well, generally, to- 
design the moving parts of the machine, and re-design 
them, until each part, the relation of each to each, and 
all as a unit, is satisfactory. After this, the frame may 
be designed to support and withstand the stresses or 
shocks of the moving parts. Oftentimes it will require 
skill to do this, but once done it is finished, whereas a 
moving part, shaped awkwardly or weakly, to accom- 
modate a frame, will be a continual source of annoyance^ 



CAMS 



The designing of Cams is taken up here because it is 
easy of mastery, interesting beyond nearly all other 
simple elements of machinery, and useful far beyond 
existing engineering practice. 

Definition. (Worcester.) *'Cam Wheel — Awheel formed so as 
to move eccentrically, and produce a reciprocating and interrupted 
motion in some other part of machinery connected with it." 

This definition, though fairly good, gives but a misty 
idea of the form or construction of a Cam. Generally 
speaking, cam motions produce the best irregular posi- 
tive motions known. They dispense w^ith expensive 
link or lever w^ork, and may easily be so designed, 
made and applied, as to produce required motions to 
the last degree of accuracy. Cams may be classified 
as follows : into Wipers and Frog Cams, Face Cams, 
Square Cams and Edge Cams. 

A Wiper is shown in Fig. i. It is used 
for operations where the power is grad- 
ually stored and suddenly expended. Trip 
hammers are often operated by such cams. 
Its object is gained by various proportions, 
but always with the same idea — a gradual 
rise, and sudden drop — the breadth of face and length 
of drop being proportioned to the work to be done. 




MECHANICAL DRAWING. 7I 

FACE CAMS. 

When a Face Cam is to be used in a machine its de- 
sign is usually left until the other moving parts and the 
points of support have been designed and located. If 
possible, the cam is placed on a shaft that is useful in 
the machine for some other operation. If this is not 
possible, then shafts are put in for the cam and cam 
lever, and the cam shaft given a positive revolution by 
gearing from the prime mover of the machine, and the 
cam designed as follows : First, about the centre F, 
(Fig. 2) draw a circle to represent the shaft and then 
one to show the cam hub. Outside these draw a third 
which shall pass through the centre of the cam-roll at 
its inside position. This position should be selected as 
near as possible to the cam hub, to save power and 
space. From the same centre draw a fourth circle 
through the centre of oscillation of the cam lever (X) . 
This last circle is divided into any convenient number 
of parts. That of Fig. 2 is divided into 33 but 32 would 
be better. From the centre selected for the lever, with 
a radius equal to its length — which is from centre of 
lever shaft to centre of cam roll — draw an arc from near 
cam centre to the circle through centre of lever shaft. 
Locate the cam roll at the intersection of this arc with 
the third circle. 

This radial arc, through the first position of the cam 
roll, is numbered i. It does not, necessarily, pass 
through the centre of the cam, though that is the best 
arrangement. Now, through each one of the remain- 
ing 32 dividing points, draw arcs like number i, with 
centres on circle through X. On these cd^cs the succes- 
sive positions of the eentre of the cam roll are platted. 
If it is desired to have a throw of two inches in one- 
quarter revolution, then, with 32 divisions, a position 



72 



MECHANICAL DRAWING. 



one-quarter inch farther from the centre of the cam, is 
taken on 8 successive arcs, when the required throw of 
two inches will have been accomplished. If, now, a 
^' rest'* is required, the points on successive arcs will be 
at the same distance from the centre of the cam shaft. 
From each one of the points as centres, just mentioned, 
a circle is drawn of the size of the cam roll, and lines 
drawn tangent to all these circles will represent the path 
in which the roll is to move. The cam of Fig. 2 has 
one-half revolution rest and then does its work in one- 
eleventh of a revolution. The rest is represented by 
making the path concentric with the cam shaft. In 




MECHANICAL DRAWING. 



73 



turning from this part of the path — just after arc 15 — 
several extra positions of the cam roll are platted, and 
again at the end of the "throw" several more. The 
object of this is to ascertain whether or not we are try- 
ing to change the direction of the path too abruptly. As 
long as each of the circles representing the roll shows a 
part of its circumference beyond the others, on tlie in- 
side line of the path, the curve is practicable ; provided ^ 
also, that the radius of curvature of the inside side of 
the path is less than that of the roll. These conditions 
must be fulfilled or the roll will not move round the 
curve. There are no very sharp curves in this cam, but 
those at 15 and 19 will illustrate the rule. From 20 to 
22 the " return " is rapid— to draw 7" from its work — and 
from 22 to I the return is made gradually and easily. A 
cam like this, with bell crank and " tucker" T^ may be 
used in a paper folding machine. 



Frtr^ 




rig.2B 



^y///////y////.^///////A 



Fig. 2 B shows something further of the construction 
of the cam and its lever. It is a section of the cam and 
roll on a line perpendicular to Z X with T left out and 
the bell-crank turned round, to show full length of L X, 
The lever should be fastened to its shaft by two set- 
screws, and the hub B should be three times as long as 
the cam roll. Fig. 2 B shows the proper form of stud 
C for the cam roll. Hub A should be of same length 
as hub B. 

This cam is designed in a wheel, but it should be un- 
derstood that only so much of it as will serve to hold the 
7 



74 MECHAN-ICA.L DRAWING. 

roll in its place and the cam on the shaft is necessary. 
It may be in form of a wheel, to prevent entanglements 
with clothing or parts of machinery. On arc 19 the 
thickness of metal outside the path is supposed to be of 
just sufficient strength to do its work. This makes the 
outside boundary of the cam. As seen in Fig. 2 B the 
luheel is a simple disc and rim. 

A '^frog " ca7?i is designed in the same way as a face 
cam. If all of this cam, outside the inside line of the 
path were cut away, the remaining — inner — part would 
be a frog cam. The cam roll for a frog cam must be 
kept against the cam by a spring or weight. 

The principle of designing face and frog cams, 7i'//// 
the lever ^ is not generally understood, but from the fore- 
going explanations it may be seen : 

1st. That if these cams were designed on radial 
lines, and the cam roll oscillated on a lever, an allow^- 
ance would have to be made for the difference of angu- 
lar position. To make this perfectly clear, design a 
cam of a quarter rest, a quarter throw, a quarter rest and 
a quarter return, by the method just described and also 
by radial lines. In the former the '' quarters" will not 
be equal ones— of 90^ — but in the latter the quarters 
will be equal. 

The conclusion must be that if the latter method of 
design is to be used the cam roll must move in a radial 
line, not in an arc of a circle. The advantage of the 
former method is that most cams do actuate a lever and 
the method plats accurately each successive position of 
its cam roll. 

2d. That as long as L X remains of same length, 
and X at same distance from F, the cam will be the 
same — to accomplish a given motion — whether A' is at 
the right or left, above or below Y. 



MECHANICAL DRAWING. 75 

3d. That if length of Z X is changed, the cam — to 
produce same results— will have to be. This is true 
whether point X is moved or not. 

4th. That, retaining X in its relative position to F, 
lengthening L X increases the power and decreases the 
speed attained at T, ^nd shortening L X produces the 
contrary effects. In both these changes the shape of the 
cam — to produce a given motion— would be of different 
form. There is no better practice for the beginner than 
to take the same requirements and design the cam under 
differing dimensions and relations of shafts. He will 
then learn of the adaptability of the method just de- 
scribed to various trying conditions and requirements. 

SQUARE CAM. 

Figures 3 and 4 are side and section views of a 
"Square Cam" and its immediate attachment, the 
'^ Yoke." This cam has often been used in sewing ma- 
chines and boot and shoe machinery. In the position 
shown the top and bottom parts are arcs of circles hav- 
ing the centre of the shaft for a centre, and are, in canr 
language, " rests." This is an unfortunate term here, 
for this cam has no rest nor gives any to its attachments. 
The essentials in drawing this cam are but two. Firsty 
there is a point on e^ch of the diagonals E F^ I H^ 
which is a centre for two arcs of circles, each forming a 
part of the outline of the cam. Second, the sum of the 
radii of these two arcs is equal to the sum of the radii of 
the two arcs of '' rest." Great care must be taken in 
drawing this cam to get the outline perfectly. The cam 
K^ Fig. 3, will, by means of its yoke, pivoted at N^ de- 
scribe with the point the " square " figure H' E I' F. 
This result is not obtained so simply by any other 
means. 

To lay out the cam, begin as in the case of the Face 



76 



MECHANICAL DRAWING. 



Cam ; draw a circle to represent the cam shaft. Through 
its centre draw two lines at right angles to each other^ 
preferably at angles of 45^, which will make four divis- 
ions of the outline of the cam. In the lowest of these 
divisions draw a quarter circle for a quarter of the cam^ 
making its radius such as will give the cam sufficient 





strength. In the opposite quarter draw another quarter 
circle, the difference between the raclii of the two quarter 
circles being the ^'' throw'' of the cam. Two horizontal 
lines may now be drawn tangent to the quarter circles, 
and two perpendicular ones also, 7?iaking a perfect square 
divided into two equal rectangles by a perpendicular through 
the ceritre of the shaft. 

The remainder of the outline of the cam may now be 
completed, with the help of previous statements, bearing 
in mind that each of the remaining arcs, of the outline 



MECHANICAL DRAWING. 77 

of the cam, must be tangent to the perpendicular sides 
of the square. 

The yoke of Fig. 3 is similar to one designed for a' 
nailing machine, the stress on it being chiefly in a ver- 
tical direction. If there were to be any considerable 
stress sidewise, the yoke should be strengthened about 
the fulcrum N. 

Fig. 4 is a section cut by a vertical plane through the 
centre of the shaft of Fig. 3. It will be easily under- 
stood, with the exception of the " sliding " block about 
the stud N^ which serves as a fulcrum. It is a square 
block bored to fit the stud. A better design for the stud 
would be one having a larger diameter through the 
block and washers, and then cut down for the threaded 
portion, at each end, and fitted with a common hex- 
agonal nut on the face. 

EDGE CAMS. 

Figures 5, 6 and 7 show an Edge Cam and methods 
used in its design. Fig. 5 shows it in connection with 
roll and lever. Fig. 6 shows the ordinary division of 
the circumference into 8 divisions of 45"' each. Fig. 
7 shows the development of the circumference with the 
path traced upon it. The largest diameter of the roll 
is used in tracing the path. It is assumed that three 
throws and three rests are required. 

The first quarter, i to 3, is a rest in the middle of the 
lever angle. In the next 8th revolution, (3 to 4), the 
lever takes its extreme " back" position. The quarter, 
4 to 6, is a rest in this extreme position. In the next 
quarter, 6 to 8, the work is supposed to be executed. 

The whole throw forward is eftected at an easy angle 
in one-quarter revolution. One sixty-fourth rest is given 
to hold the work, and seven sixty-fourths revolution re- 
turns the lever to its first position. 



78 



MECHANICAL DRAWING. 



It Will be noticed that the cam roll 
for this cam is conical. This is neces- 
sary, because that part of the roll near- 
est A must travel a much greater dis- 
tance than the end nearest the centre 
of the cam. Now that your attention is 
called to the fact, you will readily com- 
prehend this necessity ; but there are 
many cams of this kind running with 
rolls that are right cylinders simply be- 
cause their designers forgot this. In 
Fig. ^ the cam lever is shown at point 
I , with the centre line of the lever pass- 
ing through the centre of the throws of 
the cam. From 3 to 4, the roll drops 
seven-sixteenths of an inch, and from 4 
to 6, the lever makes an angle with the 
centre line of the cam, as is shown by 
the dotted line drawn from A' to the 
correct position of the roll. 

In laying out the cui^ves of the cam 
path at the ends 
of the "throws," 
much care must 
be taken not to 
have sharp an- 
gles. The cen- 
tre line is laid 
out — that is — the 
centre line of 
movement. The 
changing of this 
line from a rest 
to a throw, or the 
reverse, is always 
made in the arc 











/ 


/ 


k 


V 


\ 


v6 


\ 


n 


^ 


■i'tl 


> 




» 


»0 




V 




5^ 


\ 

\ 

i 
i_ii 1 




MECHANICAL DRAWING. 79 

of a circle. The radius of this arc must be at least one- 
eighth of an inch larger than the largest radius of the 
cam roll. This ensures a round corner for the path. It 
is better, where possible, to make this arc one-quarter 
of an inch larger than the radius of the roll. The 
change in direction of this centre line can never be over 
45° for forward or working angles. 30^ angles should 
not be exceeded. 

From this description it will be seen that the develop- 
ment is the important feature in designing this cam. 
The larger the diameter of the cam the easier the angles. 
This and all other cams should be keyed or pinned to 
the shaft. The cam levers for all cams should be fasten- 
ed with set-screws, and the correct timing of the ma- 
chine effected by slipping the lever on its shaft until the 
proper position is reached. 

LEADERS. 

When cams are to be cut, it is best to make the for- 
mer — called the "leader" in this case — much larger than 
the cam, so as to avoid inaccuracies and unsteady lines 
in the cam. The cam blank is also marked. This is 
accomplished by duplicating the development on a piece 
of thin tin, which is then fastened to the blank and the 
outline prick-punched through. Sometimes the devel* 
opment is on strong manilla paper, which is wrapped on 
the cam and pricked through. 

Edge Cams are sometimes, and Face Cams generally^ 
cast with the path in them. Face Cam patterns and 
castings are sometimes made so nicely that a cold-chisel 
and file will in a very few minutes smooth the casting 
sufficient for the roll. Such cams wear a long time. 

Usually cams are cut from leaders of double their size. 
These leaders are disks of cast iron, half an inch thick, 



8o MECHANICAL DRAWING. 

turned smoothly and emery-polished, so that the scriber 
lines may be seen easily. The leader for such a cam as 
No. 5, for example, would be a plain disk eight inches 
in diameter with the throw of seven-eighths of an inch 
and two throws of seven-sixteenths of an inch laid off 
on its edge. This, when ready for use, would resemble 
the cam but slightly, although having the same rises and 
falls on its periphery as the centre of the cam roll will 
make in traveling its path. 

The true cam in the cases of the Face Cam and the 
Edge Cam is the curve traced by the centre of the roll. 



GEARING. 



In this chapter the student may find : First, a clear 
indication of methods of drawing three forms of gear 
teeth. Second, a diagram and its description by which 
gear wheels may be proportioned ; and, third, a few 
practical hints. Several standard works have been con- 
sulted, especially the treatise published by the Brown & 
Sharpe Mfg. Co., from which are taken, by permission, 
methods for delineating single-curve and involute forms 
of teeth, and also useful formulas and explanations. 
While a larger portion of the space is used in describing 
methods used in drawing teeth than in explanation of 
the designing of gears, the student should direct his 
chief thought to the latter work, since there are several 
manufacturers who make better gear teeth than any in- 
experienced man can hope to. 

Let two cylinders, mounted on parallel axes, have 
their convex surfaces in contact. If now we turn one 
cylinder, the adhesion of its surface to the surface of the 
other will make that turn also. The surfaces touching 
each other, if there is no slip, will evidently move 
through the same distance in a given time. This sur- 
face speed is called linear velocity. Linear velocity is 
the distance a point moves in a given direction in a 
given time. These cylinders in turning about their axes 
also pass through angles whose vertices are at the axes 



82 MECHANICAL DRAWING. 

of the cylinders. The angular distance passed through 
in a given time is called angiUar velocity. If one cylin- 
der is twice as large as the other, the smaller will make 
two turns while the larger makes one, but the linear 
velocity of the cylinders is equal. 

This combination would be a very useful one in 
mechanism if we could be sure that the cylinders would 
not slip on each other. 

Let grooves be cut on the circumferences of the cyl- 
inders of a size equal to the spaces between the grooves, 
and the material taken out in making the groove placed 
on the spaces between the grooves. The spaces are 
called lands^ and the parts placed upon them addenda. 
A land and its addendum is called a tooth. A toothed 
cylinder is called a gear. Two or more gears with teeth 
interlocking are called a train. The circumference of 
the cylinders, without teeth, is called the pitch circle. 
This circle exists geometrically in every gear and is 
called the pitch circle^ or the primitive circle. In the 
study of gear wheels it is the problem to shape the teeth 
that the pitch circles will just roll on each other without 
slipping. 

The groove between two teeth is called a space. In 
cut gears the width of space and thickness of tooth at 
pitch line are equal. 

The circular pitch is the distance measured on the 
pitch line, or pitch circle, which embraces a tooth and a 
space. In cast gears the tooth is from .46 to .48 of the 
circular pitch. 

CLASSIFICATION. 

If we conceive the pitch of a pair of gears to be the 
smallest possible, we finally reduce the teeth to mere 
lines on the original pitch surfaces. These lines are 
called elements of the teeth. Gears may be classified 



MECHANICAL DRAWING. 83 

with relation to the elements of their teeth, and also 
with relation to the direction of their shafts. 

First. Spur Gears ; those gears connecting parallel 
shafts and whose tooth elements are straight. 

Second. Bevel Gears ; those gears connecting shafts 
whose axes meet when sufficiently prolonged, and the 
elements of whose teeth are straight lines. In bevel 
gears the surfaces that touch each other, without slip- 
ping, are upon cones or parts of cones, whose apexes 
are at the point where the centre of their shafts meet. 

Third. Worm Gears ; those whose axes neither meet 
nor are parallel, and the elements of whose teeth are 
helical, or screw-like. A modification of this form of 
tooth is the skew-bevel wheel, which is used in some 
cases with a smooth surface which is a zone or frustrum 
of an hyperboloid of revolution. A hyperboloid of rev- 
olution is a surface resembling a dice-box, generated by 
the revolution of a straight line around an axis from 
which it is at a constant distance, and to which it is 
inclined at a constant angle. Of modifications of the 
spur-gear we have the internal gear, the elliptical gear, 
the segment and the rack. 

We will consider Bevel Gears and Spur Gears. We 
will use the following abbreviations : 

Let D = the diameter of addendum, or ful/ size 
circle. 

\jei D' := the diameter of pitch circle. 

Let P' = the circular pitch. 

Let / = the thickness of tooth at pitch line. 

Let s = the addendum, or face of tooth. 

Let/ = the clearance. 

Lfet D' = 2 s, or working depth of tooth. 

Let Z>' -\- / = whole depth of space. 

Let JV = number of teeth in one gear. 



§4 MECHANICAL DRAWING. 

Let TT =z 3.1416, or circumference when diameter is i. 

If we multiply the diameter of any circle by tt, the 
product is the circumference of that circle. If we divide 
the circumference by tt, the quotient will be the diam- 
eter of that circle. 

The circular pitch and number of teeth in a wheel 
being given, the diameter of the wheel and size of tooth 
parts are found as follows : 

Dividing by 3.1416 = multiplying by g.^^^^ = .3183 ; 
hence, multiply the circumference of a circle by .3183 
and the product is the diameter of the circle. 

Multiply the circulai' pitch by .3183 and the product 
w^ill be the same part of the diameter of the pitch circle 
that the circular pitch is of the circumference of pitch 
circle. This part, or modulus is called a diavieter pitch. 

The diameter pitch = addendum of tooth = s. Cir- 
cular pitch multiplied by .3183 1=1 s^ or .3183 P' = s. 

The number of teeth in a wheel multiplied by a diam- 
eter pitch equals diameter of pitch circle, JVs = £>'. Add 
two to the number of teeth, multiply the sum by s and 
the product will be the whole diameter, (7V^-|-2) s = D. 
One-tenth of thickness of tooth at pitch line, equals 
amount added to bottom of space for clearance, yV = /• 
^^« = ^' = Radius of pitch circle. 

Distance between centres of two ndieels equals the sum 
of the two pitch circle radii. 

In making drawings of gears and in cutting racks, it 
is necessary to know the circular pitch in whole inches 
and the natural divisions of an inch, as one-half inch 
pitch, one-quarter inch pitch, etc., but since it is difficult 
to measure the circumference of the pitch circle and di- 
vide it into equal parts, it is much better that the diam- 
eter of a gear should be of a size conveniently measured. 

The same applies to the distance between centres. 



MECHANICAL DRAWING. 85 

Hence it is generally more convenient to assume the 
pitch in terms of the diameter. A definition of a diam- 
eter pitch and the method of obtaining it from the cir- 
cular pitch has been given. 

If the circumference of the pitch circle is divided by 
the number of the teeth in the gear, the quotient v^ill be 
the circular pitch. If the diameter of the pitch circle is 
divided bv the number of the teeth, the quotient will be 
a diameter pitch. Thus, if a gear has forty-eight teeth 
and a pitch diameter of twelve inches, the diameter pitch 
is twelve inches divided by forty-eight, or one-quarter of 
an inch. Naturally, in deciding dimensions of teeth for 
a gear, a diameter pitch of some convenient part of an 
inch is taken. 

In speaking of diameter pitch, only the denominator 
of the fraction is named. One-third of an inch diame- 
ter pitch is called 3 diametrical pitch. Diametrical pitch 
is the number of teeth to one inch of diameter of pitch 
circle. Represent this by F. Thus, one-quarter inch 
diameter pitch becomes 4 diametrical pitch, or 4 P, be- 
cause there would be four teeth on the gear to every inch 
of diameter of its pitch circle. 

The circular pitch and different parts of the teeth are 
derived from the diametrical pitch as follows : 

(i) ^-^ = P\ or 3.1416 divided by the diametrical 
pitch is equal to the circular pitch. 

(2) 7 = i^, or one inch divided by the thickness of 
one tooth equals number of teeth to one inch. 

(3) p = /, or 1.57 divided by the diametrical pitch 
gives thickness of tooth at pitch line. 

(4) p r=: Z^', or number of teeth in a gear divided 
by the diametrical pitch equals diameter of the pitch 
circle. The diameter of the pitch circle of a wheel hav- 
ing 60 teeth, 12 P^ would be, consequently, five inches. 



S6 MECHANICAL DRAWING. 

(5) —p- = Z>, or, add 2 to the number of teeth in a 
wheel and divide the sum by the diametrical pitch, and 
the quotient will be the lohole diameter of the gear or the 
diameter of the addendum circle. The diameter of gear 
blank for a gear of sixty teeth, 12 /^, would be, conse- 
quently, 5 ^2 inches. 

(6) > = P^ or number of teeth divided by diameter 
of pitch circle, in inches, gives the diametrical pitch. 

(7) -^ = P^ or add 2 to the number of teeth, divide 
by whole diameter and quotient will be diametrical pitch. 
PD' = N^ or pitch circle diameter multiplied by dia- 
metrical pitch equals number of teeth in the gear. 

(8) Formula (i) may be transposed, -^ =/^. 



SINGLE CURVE GEARS. 

Single curve teeth are so called because their working 
surfaces have but one curve, which forms both face and 
flank of tooth sides. This curve is, approximately, an 
involute. In a gear of 30 teeth or more, this curve may 
be the single arc of a circle, whose radius is one-fourth 
the radius of the pitch circle. A fillet is added at the 
bottom of the tooth, to make it stronger, equal in radius 
to one-sixth the widest part of tooth space. 

A cutter made to leave this fillet has the advantage of 
wearing longer than it would if brought up to a corner. 

In gears having less than thirty teeth this fillet is made 
the same as just given, and the sides of teeth formed 
with more than one arc, as will be shown fartheron. 

Having calculated the data of a gear of 30 teeth, \" 
circular pitch we proceed as follows : 

1. Draw pitch circle and point it ofl' into parts equal 
to one-half circular pitch. 

2. From one of these points, as at B^ (see plate 



MECHANICAL DRAWING. 



87 



Single Curve Gear) draw radius to pitch circle, and 
upon this radius describe a semicircle ; the diameter of 
this semicircle being equal to radius of pitch circle. 
Draw addendum, working depth and whole depth cir- 
cles. 

3. From the point B^ where semicircle, pitch circle, 
and outer end of radius to pitch circle meet, lay off a 




GEAR, 30 TEETH, 
^"CIRCULAR PITCH, 
P'=f'or .75" 
N=30 
P =4.1£ 
= .375" 
= .2387" 
D"= .4775" 
S^-/= .2762" 
D"+/- .5150" 
D' = 7.1610" 
D =7.7384" 



SINGLE 



GEAR. 



88 MECHANICAL DRAWING. 

distance on semicircle equal to one-fourth of the radius 
of pitch circle, shown in the figure at B A, This is laid 
off as a chord. 

4. Through this new point A, upon the semicircle, 
draw a circle concentric to pitch circle. This last is 
called the base circle^ and is the one for centres of tooth 
arcs. In a certain system of single curve gears, the 
diameter of this circle is .968 of the diameter of pitch 
circle. 

5. With dividers set to one-quarter of the radius of 
pitch circle, draw arcs forming sides of teeth, placing 
one point of the dividers in the base circle and with the 
other point describing an arc through a point in the 
pitch circle that w^as made in laying off the parts equal 
to one-half the circular pitch. Thus with A as centre, 
an arc is drawn through B. 

6. With dividers set to one-sixth of the widest part 
of tooth space, draw the fillets for strengthening teeth 
at the roots. These fillet arcs should just join the whole 
depth circle to the sides of teeth already described. 
Single curve or involute gears are the only gears that 
will run at varying distances of axes, and transmit un- 
varying velocity. This peculiarity makes involute gears 
especially valuable for driving rolls or any rotating pieces, 
the distance between whose axes is likely to be changed. 

The assertion that gears crowd harder on bearings 
when of involute than when of other forms of teeth, has 
not been proved in actual practice. 

It is an excellent practice to cut out the drawings of a 
pair of gears, that have been made on Bristol-board, and 
test their accuracy in running together. 



MECHANICAL DRAWING. 89 

RACK TO MESH WITH SINGLE CURVE 
GEARS HAVING 30 TEETH AND OVER. 

This gear is made precisely the same as the one last 
described. It makes no difference in which direction 
the construction radius is drawn, so far as obtaining 
form of teeth and making gear is concerned. Here the 
radius is drawn perpendicularly to pitch line of rack 
and through one of the tooth sides, B. A semicircle is 
drawn on each side of the radius of the pitch circle. 
The points A and A ' are each one-fourth the radius of 
the pitch circle distant from point B^ and correspond to 
the point A in the last figure. In this construction draw 
the lines BA and BA '. These lines form angles of 75^° 
(degrees) with radius BO. Lines BA and BA' are 
called lines of pressure. The sides of the rack teeth are 
made perpendicular to these lines. 

A Back is a straight piece having teeth to mesh with 
a gear. A rack may be considered as a gear of infinitely 
long radius. The circumference of a circle approaches 
a straight line as the radius increases, and when the 
radius is infinitely long any finite part of the circum- 
ference is a straight line. The pitch line of a rack, 
then, is a straight line just touching the pitch circle of a 
gear meshing with the rack. The thickness of teeth, 
addendum, and depth of teeth below pitch line are cal- 
culated in the same manner as for a wheel. 

A rack to mesh with a single curve gear of 30 teeth 
or more is drawn as follows : 

1. Draw straight pitch line of rack; also draw ad- 
dendum line, working depth line and whole depth line, 
each parallel to the pitch line (see figure) . 

2. Point oft' the pitcli line into parts equal to one- 
half the circular pitch, or = t. 



90 



MECHANICAL DRAWING. 



Fig.r 




RACK TO MESH WITH SINGLE CURVE GEAR 
HAVING 30 TEETH AND OVER. 



MECHANICAL DRAWING. 9I 

3. Through these points draw lines at an angle of 
75i^ with pitch lines, alternate lines inclined in opposite 
directions. The left-hand side of each rack tooth is 
perpendicular to the line B A, The right-hand side of 
each rack tooth is perpendicular to the line B A' , 

4. Add fillets at bottoms of teeth equal to one-sixth 
of the width' of spaces between the rack teeth at the ad- 
dendum line. 

ANGLES FOR RACK TEETH. 

To get the proper angle for rack teeth, draw a semi- 
circle on a line A B. With centre A^ and radius equal 
to one-quarter of A B draw radius, cutting semi-circle 
at C. A straight line through A C will form an angle of 
75|^ with the line A B. 

To get the angle for sides of a tool for planing out 
rack teeth proceed as follows : On line O B describe a 
circle. From B lay off on the circumference chords B A 
and B C, each equal to one-fourth of O B. Angle A O C 
is 29^ — the proper angle for the point of the tool. Make 
end of rack tool .31 of circular pitch, and then round 
the corners of the tool to leave fillets at the bottom of 
rack teeth. Thus, if the circular pitch of a rack is i^", 
and we multiply it by .31, the product, .465", will be the 
width of tool at end, for rack of this pitch, before the 
corners are taken of^\ 

GEARS AND RACKS TO MESH WITH GEARS HAVING 
LESS THAN 30 TEETH. 

The construction of the rack is similar to that just 
described. (See Figure Gear 2 P., 12 teeth in mesh 
with rack) . 

The curve on face of tooth, or that part outside of 
pitch circle, may be constructed as for a gear having 30 



92 MECHANICAL DRAWING. 

teeth or more, but \\\^ flanks^ or curve of tooth inside of 
pitch circle, are drawn as follows : In gears of 12 and 
13 teeth the flanks are parallel for a distance, inside the 
pitch circle, of one-third of a diameter pitch (^/^ ^) gears 
of 14, 15 and 16 teeth, one-fifth s; 17 to 20 teeth, one- 
sixth s. In gears of more than 20 teeth the parallel con- 
struction is omitted. 

The involute tooth of this gear of 12 teeth is drawn as 
follows : Draw three or four tangents to the base circle, 
iV jj\ kk' n\ letting the points of tangency on the base 
circle i\ j\ k\ /', be one-fourth of the circular pitch apart ; 
the first point, /', being distant from / one-fourth of radius 
of pitch circle. With dividers set to one-fourth the radius 
of pitch circle, placing one point on /', draw the arc 
a' ij; with one point on/', radius^*', drawy'y^y with one 
point on k' draw kl. Should the addendum circle be 
outside of / the tooth side may be completed with the 
last radius //'. 

The arcs a ' i, ij\ Jk, kl^ together form a very close ap- 
proximation to a true involute from the base circle ij'k'T . 
The exact involute for gear teeth is the curve made by 
the end of a band when unwound from a cylinder of the 
same diameter as the base circle. 

With diameter equal to the distance between the ends 
of two adjacent involutes, where they meet the base cir- 
cle, draw a circle about centre of gear. Lines from 
these points tangent to the ciicle form part of the flanks 
of teeth. From the whole depth circle, draw fillets 
with radius equal to J^ widest tooth space. These will 
butt into the parallel lines about ^ s from the base 
circle. 

This method is conventional, depending upon the 
judgment of the designer, to eflect the object to have 
spaces as wide as practicable just inside base circle and 
then strengthen flank with as large a fillet as will clear 



94 MECHANICAL DRAWING. 

addenda of any gear. If flanks in any gear will clear 
addenda of a rack, they will clear addenda of any other 
gear, except internal gears. An internal gear is one 
having teeth on the inner side of a rim or ring. 

The foregoing operation of drawing tooth sides, al- 
though tedious in description, is very easy of practical 
application. 

The faces of teeth of rack are rounded oflf by an arc 
or radius of i ^ pitch, with centre in working depth 
line. 

BEVEL GEAR BLANKS. 

The pitch of Bevel Gears is always figured at the 
largest pitch diameter. 

Most Bevel Gears connect shafts that are at right 
angles to each other. The following directions apply 
to any angle, but the sketch is made with axes at right 
angles. 

Having decided upon the pitch, numbers of teeth and 
angle of shafts : (The sketch is made for gears i.i and 
2.2 inches diameter.) 

Draw axes AOB, COD, Fig. i8. At a distance from 
AOB, equal to one-half the diameter of the gear, dra'w 
a line parallel to AOB. At a distance from COD, equal 
to one-half the diameter of the pinion, draw a line par- 
allel to COD. From the point ;/, where these two 
parallels meet, draw perpendiculars to AOB and COD, 
On these perpendiculars lay oft' the pitch diameters, // 
of the gear, and ;;/// of the pinion, the point / being com- 
mon. From y, ;/ and ;;/ draw lines to O These lines 
give size and shape of pitch cones, and are called Cone 
Fitch Lines. 

Through points ;;/, / and y, draw lines mx, iy and jz 
perpendicular to Cone Pitch Lines. 

On these lines, from cone pitch line, lay off' distances 



MECHANICAL DRAWING. 



95 



for addenda, working depth and whole depth of teeth. 
From the points so obtained, draw lines to the centre O. 
These lines give the height of teeth above Cone Pitch 
Lines, and the whole and working depths of teeth. 




The teeth become smaller as they approach O and be- 
come nothing at that point. It is quite as v^ell never to 
have the length or face of teeth, imn^ longer than one- 
third the distance Om^ nor more than two and a half 
times the circular pitch. 

Having decided upon the length of face, draw limit- 
ing lines 7n'x\ i'j' and j'^z'. 

We have now the outline of section of gears through 
their axes. A straight line drawn through the largest 
diameter of the teeth, perpendicular to axis of the gear, 
is called the largest diamete?^ In practice, these diam- 
eters are obtained hy measuring the drawing. 

To obtain data for teeth, w^e need only make drawing 
of section of one-half of each gear. 

We first draw centre lines AO, BO and lines ^i;// and 
€d, then gear blank lines as in the case just described. 
(See Fig. 17.) 

To obtain shape of teeth in bevel gears, we do not lay 
them off on pitch circles in same way as in spur gears. 




S = .200" 

D'= .400' 

8+/= .231" 

DM-/ - .431" 



D"'= .266' 
S'+/ =.165' 
D"'+/ =.298' 



BEVEL GEARS. FORM AND SIZE OF TEETH. 



MECHANICAL DRAWING. 9^ 

A line running from a point on cone pitch line to 
'Centre line of a bevel gear, perpendicular to this cone 
pitch line, is the radius for circle upon which to draw 
outlines of teeth at this point. 

Hence Ac is the geometrical pitch circle radius, for 
large end of teeth, and AW the geometrical pitch radius 
for small end of teeth of wheel. To avoid confusion, 
the distance A'c' is transferred to Ac' , 

For the pinion we have the geometrical pitch circle 
radius Be for large end of teeth, and the radius B'c for 
small end of teeth. Transfer distance B'c' to line 
Be"''.'' 

About A^ draw arc cn7n, and upon it lay off spaces 
equal to the thickness of tooth at pitch line, and draw 
outlines of teeth as previously described. 

We have now the shape of teeth at large end, repeat 
this operation with radius Be about B, and we have form 
of teeth, at large end of pinion. 

Upon arc of radius A'e' we get shape of teeth of 
small end of gear, and upon arc of radius B'e' we get 
shape of teeth at small end of pinion. 

The sizes of tooth parts at small end may be taken 
directly from the diagram, or they may be calculated as 
follows : 

Dividing the distance Oe'^ which, for example, may 
be 2 inches, by Oe, which may be three inches, we get 
Yz or .666 for a ratio. Multiplying outside sizes by 
.666, we get the corresponding inside sizes. Thickness 
of teeth at outside being .314 inchj fi of it gives us .209 
inch as thickness of teeth inside. 

When cutting bevel gears with rotary cutters, tlie 
angle of cutter head is set the same as angle of working 
depth ; thus : To cut the gear we have the cutter travel 



* Tredgold's method from Rankine, App. Mech. p. 448. 
9 



98 



MECHANICAL DRAWING. 



in the direction Op. The angle AOp is called the " cut- 
ting angle," being measured from the axis of the gear. 
In this method the angle of face of pinion is the same 
as cutting angle of gear, and face angle of gear is the 
cutting angle of pinion, and clearance is the same inside 
as outside. 

EPICYCLOIDAL TEETH. 



An epicycloid is 
" a curved line gen- 
erated by a point in 
the circumference of 
a circle, which rolls ^ 
on the circumference ^ 
of another circle, S 
either internally or ^s 
externally." — (Wor- 
cester.) 

Hence an epicy- 
cloidal tooth has parts 
of epicycloids for the 
curves of its faces. 

In the sketch, hav- 
ing determined on 
one-inch pitch, and 
found radius of roll- 
ing circle from dia- 
gram, the point ^was 
selected for a starting 
point, and the centre 
of the rolling circle 
having its centre at 
r' rolled on the pitch 
circle through four 
stations as shown, 




MECHANICAL DRAWING. 99 

and the epicycloid traced by o found to be ox^ for the 
face of the tooth, and in the same manner, the epicycloid 
ox developed for the flank of the tooth. Four positions 
of the centre of the rolling circle, and four of the point 
are given in each case. 

By the diagram, thickness of tooth, / = .48 pitch. 
Length of tooth, I = ,^ F. Three-tenths of this dis- 
tance out from the pitch circle determines a point in 
the addendum circle, and four-tenths pitch in from the 
pitch circle gives a point in the w^hole depth circle. 
This allows -^-^ P for clearance. 

The curves obtained for one side of the tooth may now 
be reversed at a distance of .48 P^ and we have the out- 
line of a tooth that may be duplicated around the wheel. 

"It is considered desirable by millwrights, with a 
view to the preservation of the uniformity of the shape 
of the teeth of a pair of wheels, that each tooth in one 
wheel should work with as many different teeth in the. 
other wheel as possible. 

" They, therefore, study to make the numbers of teeth' 
in each pair of wheels which work together, such as to 
be prime to each other, or to have their greatest com- 
mon divisor as small as is possible consistently with the 
purposes of the machine. 

" The smallest number of teeth which it is practicable 
to give a pinion is regulated by the principle, that in 
order that the communication of motion from one wheel 
to another may be continuous, at least one pair of teeth 
should always be in action ; and that in order to provide 
for the contingency of a tooth breaking, a second pair, at 
least, should be in action also."* 

The least number of teeth that can usually be em- 
ployed is as follows : 

* Rankine. 



lOO MECHANICAI. DRAWING. 

Involute teeth, 25; epicycloidal teeth, 12; cylindrical 
teeth, or staves, 6. 

The Arc of Contact on the pitch lines is the 
length of that portion of the pitcJi lines w^hich passes 
the pitch point during the action of one pair of teeth ; 
and in order that two pairs of teeth, at least, may be in 
action at each instant, its length should be double the 
pitch. It is divided into two parts, the arc of approach 
and the arc of recess. In order that the teeth may be of 
length sufficient to give the required duration of contact, 
the distance moved over by the point on the pitch line, 
during the rolling of a rolling curve to describe the face 
and flank of a tooth, must be, in all, equal to the length 
of the required arc of contact. 

Link of Pressure. When one body presses against 
another, not attached to it, the tendency to move the 
second body is in the direction of the perpendicular at 
point of contact. 

This perpendicular is called the line of pressure. The 
angle that this line makes w ith the path of the impelling 
piece is called the a^igle of pressure. 

In the case of gearing, the line of pressure makes an 
angle with the line of centres of 75^ to 78^. 

PROPORTIONS OF GEAR WHEELS. 

Much of the study and w^ork on gears by engineers 
and manufacturers has been devoted to the improvement 
of the shape of the teeth of gear w^heels, and a large 
part of the illustrative chart is likewise occupied with 
representations of some of the most important of the 
results of this study and work. A matter often left hap- 
hazard and "rule of thumb" design is the proportions 
of gear wheels. 

In the sketch representing epicycloidal teeth, the de- 



MECHANICAL DRAWING. lOI 

sign of a gear wheel is completed and the proportions 
so graphically represented as to enable a student with 
very small labor to design any gear. The proportions 
of a gear of less than 12 inch diameter are of compara- 
tively small interest to the draughtsman or designer since 
they have been so often designed and manufactured that 
fairly perfect ones may be obtained of a dozen different 
manufacturers. 

When, however, we need a gear wheel of from 2 to 
10 feet diameter, we are, for various reasons, inclined to 
bestow considerable care on its design. 

The proportions given in the sketch were compiled 
from the statements of three authors, and modified some- 
what by personal experience. The pitch of the gear is 
here made the basis of all dimensions. 

As to the relative strength of the different parts of a 
gear wheel, there is a wide difference of opinion ; some 
holding that the teeth should be the weakest part and 
others contending that all parts should be equally strong 
— the latter having in mind the principle which the 
Deacon had when he built his celebrated '' one boss 
shay." 

TEETH OF GEAR WEELS. 

There are at least two good reasons why the teeth 
should be made the weakest part of a gear. The teeth 
are the smallest part of the gear, and in falling — after 
having been broken —are least likely to damage the ma- 
chine of which the gear is a part. Also, if but a few 
teeth or cogs are broken out, they may be easily replaced 
by pins with small loss of time — after which a new and 
better gear may be substituted. 

The value of a tooth in transmitting power is a sub- 
ject of great importance in this study of gears. 
♦9 



I02 MECHANICAL DRAWING. 

The following formulas for the strength of teeth are 
from Thomas Box's Practical Treatise on Mill Gearing. 
The conclusions drawn from them and also the com- 
parison of the different forms of gear teeth, which fol- 
lows, are by Mr. Geo. B. Grant, a well-known authority 
on the subject of Gearing. 

STRENGTH OF A TOOTH. 

For worm gears, crane gears, and slow moving gears 
in general, we have to consider only the dead weight 
that the tooth can lift with safety. 

If we use a factor of safety of lo, we can use the 
formula JV = 350 c. /. — in which IV is the weight to 
be lifted, c is the circular pitch, and / the face, both in 
inches. For wooden cogs, substitute 120 for 350 in 
this formula. 

When the pinion is large enough to insure that two 
teeth shall always be in fair contact, the load, as found 
by this rule, may be doubled. 

Example. A cast-iron gear of 3 in. circular pitch, 6 
in. face, will lift 

W = 350 X 3 X 6 = 6,300 lbs. 

HORSE POWER OF A GEAR. 

For very low speeds we may use the formula 
IfJ^ for low speed = -0037 ^^> ^^ ^ /> 

in which d is the pitch diameter, c the circular pitch, 
and / the face, all in inches, and // is the number of 
revolutions per minute. 

The horse power of a gear 3 ft. in diameter, 3 in., 
pitch and 10 in. face, at 8 revolutions per minute, is 

If I" = .0037 X 36 X 8 X 3 X 10 = 33 



MECHANICAL DRAWING. I03 

For ordincDj or high speeds we must use the formula 
HP^ ,012 c'f VJTt, 

When in doubt as to whether a given speed is to be 
considered high or low, compute the horse power by 
both formulae and use the smaller result. For bevel 
gears the same rule will apply, if we use the pitch diam- 
eter and the pitch at centre of face. 

The rules given above for the horse power of gears 
apply only to cast gears. One of the chief sources of 
weakness in a cast gear is that the continual pounding 
of the teeth on each other crystalizes the metal so that 
its strength is greatly decreased long before it is worn 
out. There are no recorded tests on the horse power of 
cut gears, and, consequently, we can only proceed by 
judgment and inference. Thus w^e may say that the 
weakening source — pounding — of the cast gear is absent 
with the cut gear, and from that point of view the latter 
would be stronger. In the absence of proof to the con- 
trary, we may assume that the rule that applies to cast 
gears for slow speeds, where impact need not be con- 
sidered, can safely be applied at higher speeds to cut 
gears where there is no impact to be allowed for ; and 
we have the formula : 

Horse power of cut gears at ordinary speeds = .0037 
dncf ; or, to be within bounds of safety, -^"^^^^^^-^ 

A COMPARISON OF EPICYCLOIDAL WITH 
INVOLUTE TEETH. 

This matter might easily cover many pages, but is 
condensed to the following points : 

Adjustability. Involute — single curve— teeth alone 
can possess the remarkable and practically invaluable 
property, that they are not confined to any fixed radial 



I04 MECHANICAL DRAWING. 

position with respect to each other, for, as long as one 
pair of teeth remains in action, until the next pair is 
in position, the perfect uniformity of the action of the 
curve is not impaired. 

Epicycloidal teeth must be put exactly in place and 
kept there, and the least variation in position, from bad 
workmanship in mounting, or by wear or alteration of 
the bearings in use, will destroy the uniformity of the 
motion they transmit. 

Uniformity. The direct force exerted by involute 
teeth on each other, is exactly uniform, both in direction 
and amount, and this property insures a uniform wear- 
ifig action of the teeth, a nearly uniform thrust on the 
shaft bearings, and a steadiness and smoothness of action 
that cannot be claimed for epicycloidal teeth under any 
circumstances. The direct pressure acting between 
epicycloidal teeth is variable in amount afid very vari- 
able in direction, and consequently the friction and 
wearing action between the teeth, as well as the thrust 
on the bearings, is variable between wide limits. 

Friction. This measure is always in favor of the 
involute, although the advantage is usually claimed for 
the epicycloid, both as to maximum and average values, 
and as this is an important point, it should have great 
weight in deciding between the two forms of teeth, for 
the element of friction is of chief importance in deter- 
mining the life of a gear in continual and heavy service. 

Thrust on Bearings. Here the advantage is with 
the epicycloidal tooth, but not by a large amount, and is 
not a matter of first consequence. 

Strength. The greatest strain comes at the root of 
the tooth, and as the involute tooth spreads more than 
the epicycloidal tooth, it is stronger at that point. 



MECHANICAL DRAWING. I05 



GEAR DESIGNING. 



After the cross-sectional area of the tooth is deter- 
mined on, the proportions for the rest of the gear are 
easily arrived at by looking at the sketch and accom- 
panying diagrams. 

In the right hand diagram, the vertical column of 
figures indicate circular pitch. 

The horizontal lines are simply divisions of the pitch — 
or pitches. 

The inclined lines show the variations of wheel di- 
mensions corresponding to the pitch. Thus, small r, 
the radius of rolling circle, for a 4 inch pitch gear is ^ 
of 4 inches, or 31^ inches; for 3 incii pitch, r is 2S/^ 
inches; for 3 inch pitch, r is i^ inches, and for i inch 
pitch, r \^ 1/^ of an inch. 

This matter of the size of the rolling circle is a very 
important one. Its size may be increased until the 
flanks of the teeth are straight radial lines, or decreased 
until the face of the tooth is an arc of as small a circle as 
the rolling circle itself, /. e.^ an epicycloid which nearly 
coincides with an arc of a circle equal to the rolling 
circle. 

Looking now for the thickness of rim, we find that it 
is given in the left hand diagram as <^ = i^ inch + .4 P^ 
and this for 4 inches P is 1.735 inches, or about i^ 
inches, for 3 inches pitch d is i^, for 2 inches \\ of an 
inch, and \\ of an inch for i inch pitch. This rim is in- 
creased from the edge or side of the wheel, toward the 
centre, until it is 1.2 ^ thick, where it is reinforced by a 
central rib d wide and d thick. This rim has had its 
teeth stripped from it, and therefore is strong enough, 
though it looks light. 

For computing the strength of arms we must have the 
pitch diameter and width of face of wheel given. De- 






CO 

CO 


— o 




[^ ^ PR0PQRTI0N5 OF GEAR WHEELS 



MECHANICAL DRAWING. IO7 

noting the face of the wheel by b, half the pitch diameter 
by R, and the required depth of the arm at the hub by h^ 
the following formulae for arms are considered good : 

For 4 arms h^=^ .61 V]^ 
For 6 arms /^ = .5 V^ 
For 8 arms h = .46 V^^ 
For 10 arms /i = .443 V^^ 
For 1 2 arms /i = .438 F^ 

The depth of the arm at the rim should be ^ of the 
hub depth. 

The term depth is here used to denote the dimension 
of the arm in a plane at right angles to the axis of rota- 
tion. The arm is d thick. 

To strengthen the arm against side thrust and twisting, 
a rib a is put on each side of the arm, nearly as wide as 
the face of the gear. Its thickness is .7 ^. 

The fillet between the arms at the hub should never 
be less than 5^ d deep. 

If the rib a is not used, the thickness of the arm should 
be increased to 1.2 d. The thickness, IV, of the hub 
should be 4|f inches for a lo-inch shaft and J^ an inch 
less for each inch decrease in diameter of the shaft down 
to a three-inch shaft, where W is ijV inches. For a 
2-inch shaft IV is '^ of an inch, and ^ inch for a i-inch 
shaft. 

These explanations, with the diagrams and previous 
designs of tooth forms, should enable the student to 
design gear wheels correctly and with ease. 

It may be well to add here that no amount of " book 
wisdom '' will supply a want of practical knowledge of 
the subject in hand, nor will any amount of " finger 
wisdom " enable a mechanic to design the gear he fault- 
lessly makes. 



MECHANICAL DRAWING. IO9 

Experience, judgment and " common sense," as it is 
called — though it is rather uncommon — are necessities 
to the designer. These coupled w^ith " book wisdom" 
ought to make a good designer. If in addition to these 
qualities he has plenty of " finger wisdom " and a natu- 
ral mechanical ability, his designs should be as nearly 
perfect as our civilization demands. 



10 



STRENGTH OF MATERIALS. 



For the student's convenience, a few tables, diagrams 
and alphabets are here provided. 

Strength is the resistance a body opposes to a perma- 
nent separation of its component parts. Beams of the 
same material vary greatly in strength, and they some- 
times break under one-fourth the load corresponding to 
the figures given as their breaking strength. A large 
factor of safety is hence advisable. A solid cylinder 
varies in strength as the cube of its diameter. The 
formula for this case becomes, where fixed at one end 
and loaded at the other. 

Load = 

1.7 X 6 X /. 

Where R = stress at the breaking point, d the dia ni- 
ter and / the length. If the cyhnder be uniformly 
loaded it will carry twice as much load ; supported at 
the ends, and loaded in the middle. Load becomes 
quadrupled ; supported at both ends and uniformly 
loaded, it is 8 times as great. A beam supported at 
one end and fixed at the other, and loaded uniformly, 
has the same strength as the last case, as has also a 
beam fixed at both ends and loaded in the middle. 
When fixed at both ends and uniformly loaded, the 
value of Load is twelve times as great as in the first 
of the preceding cases. 

The last statement of the rehitive strength of beams 
difterently placed is correct for all solid beams. A 



MECHANICAL DRAWING. Ill 

wooden beam of triangular section, supported at both 
ends, is about one-sixth stronger with its base upward 
than with its base downward. The strongest beam of 
rectangular section that may be cut from a round log has 
a depth proportioned to its breadth as 7 to 5. Such a 
beam is ten per cent, stronger than the largest square 
beam that may be cut from the same log. The strength 
of any beam, of whatever material, varies dif-ectly as its 
breadth and as the square of its depth. This fact is 
easily remembered in the formula bd^. Hence the 
transverse strength of a 2 by 6 beam placed edge up is 
greater than that of a 3 by 6 beam placed side up, in the 
ratio of 73 to 54. 

The transverse strength of round iron and steel may 
be taken as six-tenths the strength of bars of square 
section having their sides equal to the diameter of the 
former. A hollow cylinder has a strength exceeding 
that of a solid cylinder of the same length, weight and 
volume. Triangular beams of cast iron when the edge 
resists compression, and their resistance becomes a max- 
imum when the shape of section and the ratio of tensile 
strength to resistance in compression, are so related that 
the beam, when on the point of rupture, is equally liable 
to break by yielding to either force. The best form for 
a cast iron '' J-beam" is one in which the area of the 
lower "flange" is six times that of the upper — since 
that is about the ratio of the tensile and compressive 
strengths of cast iron. 

In general, extending the extreme portions of the sec- 
tion where stresses become greatest, and restricting the in- 
termediate part, or the "web," to the size needed to hold 
the other portions in proper relative positions, will produce 
forms of beams of greater strength, with a given weight 
of material, than can be obtained in the cases of rectan- 
gular, circular or other simple forms of section. Where 



112 



MECHANICAL DRAWING. 



the metal has equal strength to resist tension and com~ 
pression, the top and bottom flanges should be of equal 
size. This constitutes the Tredgold ^' J-beam," usually 
made of wrought iron. In many cases the form of sec- 
tion is determined by convenience, in making up. Thus 
columns are made up of " L/' '^ U/' "I'* or '' X 
beams." 

STRENGTH OF MATERIALS. 



Materials. 



Water \ 62 . 5 

Wro't Brass 5^3-^ 

" Copper 543-6 

" Iron 481 

" ' * Rope ....... 

Cast Iron I 444 

Cast Steel ; 494 

Chrome Steel 'j 492 

Plate Steel 487 

Tin, Cast 460 

Zinc ' 435 

Phosphor-Bronze, | 

Wire |. . . . 

Phosphor-Bronze, | 

Cast 

Manganese Bronze .... 
Aluminum Bronze. .... 

Ash Wood 47 

Cedar ' 35 

Oak 53.75 

Pine, White 34-62 

Pine, Yellow 33. 

Spruce 31-25 

Brick 102 

Glass 169 

Granite 165 

Leather 63 

Limestone 197 

Marble 167. 

Mortar j 98 

Slate 1178 

Rope, Manilla .... I ... . 
Rope, Hemp I . . . . 



035 
297 
314 
270 



257 






Strength — per sq. in. — lbs. 



I 

8.21 
8.69 

7.8 



7.7 

7-9 



bo 

G 



7.4 

7 



30,000 1 00000 

34,000 7,500 
50,000 50,000; 75,000 

90,000 

25,000 16,500 125000 

88,600 295000 

171000^ 
iioooo 80,000 



700 

30,000' 30, 000 
54,000 



675 
650 



5,000 
6,000 

.1110000 , 



40,000 
15,500: . 
i6,ooO| 



800 



200 



751!- 
56H 



4611. 



70,000 : 
95,000 : 
14,000; 
11.400! 



40,000 

IIOOOO 

130000 
8,000 
5,90o| 

87M I i3,6oo| 6, lOO: 

5,775 
8,200 
5,950 
2,000 
30,000 
15,300 



168 



75' 



75' 



11,800 

11,400 

10,300 

300 

8,000 

10,000 

350 

2,000 

9,000 

50 

12,000 

9,000 

15,000 



230 
130 



26 



3,065 

15,000 

120 






24,000 



90,000 



MECHANICAL DRAWING. 



113 



FACTOR OF SAFETY. 

The table of '^Strength of Materials " given in this 
book is composed of figures indicating the ultimate 
strengths of the materials. Obviously, materials in 
structures are never intended to be subjected to these 
strains. By great numbers of tests under a great variety 
of conditions ''factors of safety" have been arrived at, 
by w^hich the ultimate strengths are divided, the results 
being the '' safe w^orking loads." These factors are much 
greater under moving or "live" than under steady or 
"dead" loads, and vary w^ith the character of the ma- 
terial used. 

The factor of safety for building stone should never 
be less than ten, and sometimes a far higher value is 
adopted. It should be remembered that the crushing 
strength of the mortar or cement used as building ma- 
terial is often used rather than that of the stone. For 
machinery the factor is usually 6 or 8 ; for structures 
erected by the civil engineer from 5 to 6. The follow- 
ing may be taken as minimum values for the factor of 
safety. The figures given for materials subjected to 
shocks are approximate : 



Material. 



Copper and other soft metals and 

alloys 

Brass, brittle metals and alloys 

Wro't iron and soft steel 

Tool and machine steel 

Cast iron 

Building stone 

l^uilding timber 

1 .eather 



Load. 



'Dead.' 



5 
4 
3 
3 
4 
10 

5 



" Live. 



10 
7 to 10 



Shock. 



10 
10 to 15 

8 

9 
10 to 15 



*10 



114 



MECHi\NICAL DRAWING. 



HORSE POWER OF SHAFTING. 

The diagram is one established by J. T. Henthorn^ 
M. E., to determine the size of good hammered iron 
shafting necessary to transmit certain powers. The dia- 
gram is constructed from the formulas 

'^56 X HP 



D-. 
56 '' 



^, 



which transposed, is 



HP, or agam, is P = ^ — ^-^ 



To use this diagram, supposing that it is desired to 
drive 900 horse power at 304 revolutions per minute : 

REVOLUTIONS OF SHAFT PER MINUTE. 




MECHANICAL DRAWING. II5 

following the horizontal line representing power and the 
perpendicular line for revolutions per minute, it is found 
the nearest diagonal line is one representing 5^' in 
diameter. Or, supposing there is a shaft "j}^" in diam- 
eter and its speed is 170 revolutions per minute, and it 
is required to find the horse power v/hich it will safely 
transmit. Running along on the horizontal line of 170 
revolutions per minute until the diagonal is reached, 
representing 71^" in diameter, at this intersection is 
found a perpendicular representing the horse power, 
which is 1,155. 

STRENGTH OF LEATHER BELTING. 

The width of the belts should always be a little less 
than the face of the pulley ; both are to be determined 
by the power to be transmitted and the velocity of move- 
ment. For light work a single thickness only is neces- 
sary, but for belts from prime movers and in other places- 
where great power is to be transmitted, double belts are 
used. 

For single belts embracing half the pulley, with a 
velocity of 600 ft. per minute, one horse power can be 
transmitted for each inch in width of belt, with a max- 
imum stress on the belt of 50 pounds and pressure on the 
journals of about 85 pounds per inch of width of belt. 

J. T. Henthorn, M. E., has given the following for- 
mula for the strength of double belts, per inch in width, 
in which D is the diameter of the pulley in feet, 7? the 
revolutions per minute, and H. P. the horse power : 

^X ^^^ ^^ p 
450 

This formula gives .7 //. P. for a belt one inch wide 
running on a one foot (diam.) pulley at 100 revolutions 
per minute, double that for the same belt on a 2 ft. pul- 
ley at 100 revolutions, triple on a 3 ft. pulley, etc. 



ii6 



MECHANICAL DRAWING. 



HORSE POWER PER INXH OF WIDTH. 
2 3 4 5 6 7 8 9 10 II 12 13 14 15 



50 
60 

70. 
So- 
90. 

100 . 

no 

120 
140 

ISO 
160 
170 
180. 
190 
200 
210 
220 
230 
240 
250 
260 
270 
2S0 
290 
300 
310 
320 
330 
340 
350 





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\ 










\ 


\ 


\ 




\ 


\ 


\ 


\ 


s 


\ 










\ 


\ 




\ 


\ 


\ 


\ 


\ 


\ 


' V 


1 




1 


\ 


\ 




\ 


\ 




\ 


\ 


\ 


s, 


\ 






I 


\ 




\ 


\ 




\ 


\ 


N| 


V 


\ 


V 








\ 




\ 


\ 




\ 


\ 


\ 


\ 




\, 




L 








\ 




[ 


\ 




\ 




\ 


\ 




' 






\ 


\ 


\ 


\ 




\ 




\ 


\ 












\ 




\ 


\ 


I 


\ 




\j 




\ 










\ 




\ 




\ 




\ 




\ 


\ 










\ 




\ 




\ 




\ 




\ 


















\ 


\ 


\ 




V 


> 


\ 
















\ 




\ 




\ 




\ 












\ 




\ 




\ 




\ 
















\ 




\ 






\ 




\ 














\ 






\ 




\ 




\ 




















\ 




\ 






\ 














\ 




\ 






\ 




\ 



(U;^^Diameter of Pulleys shown by Diagonals. 

The diagram made from this formula is used as fol- 
lows : To find the horse power that can be transmitted 
bv a twelve inch belt on a nine feet pulley running at 190 



MECHANICAL DRAWING. II7 

revolutions per minute : Following the line represent- 
ing 190 revolutions and the line representing the 9 feet 
pulley, we find that they meet on the perpendicular num- 
bered 12. This means that for this pulley at this speed 
each inch width of belt will transmit 12 horse power; 
hence a 12-inch belt will transmit 12 x 12 == 144 horse 
power. To find the width of belt necessary to transmit 
50 H. 7^. on a 4 ft. pulley running at 210 revolutions : 

Inspection of the diagram shows that a belt i inch 
wide on this pulley at this speed will transmit 6 H. P.^ 
whence (50 -i- 6) a belt 8^ inches wide will be neces- 
sary to transmit 50 H. P. on this pulley at this speed. 



iiS 



MECHANICAL DRAWING. 



1 


1 


05 <oi^Cjv-«^«v5^£v.(>v ^J^^.VjC^tJ^'^cwj 


•^^ »;5 ^ ^^ ^ ^ ^ i^ ^ cs. -^ ^ bo <-;^ Q^j^ 


^ 
^ 




i? 

1 


^ ^ < Qi <:\2 <t) -;^ ^/O 'O ^_ ^ I^ ^ CS 0^ 








« 
1 

C 


^ 
^ 




'^ 


-->^'^c>^^^^^->^^S;^^ 




U-A_-^r 



HANDLE TABLE 

DIMENSIONS IN INCHES. 



A 


B 


c 


D 


E 


1 
! F 

1 

! 


G 


H 


R 


1 


i 


.2 


1 

8 


t\ 


1 

1 .0333 


3 


.02 


A 


li 


tV 


.25 


^\ 


H 


1 .0416 


i 


.025 


.5 


li 


1 


.3 


vv 


A 


! .05 


^V 


.03 


.6 


2 


i 


A 


1 

4 


2. 

8 


.0666 


tV 


.04 


.8 


2i 


f 


.5 


tV 


H 


.0833 


^V 


.05 


1.0 


3 


i 


.6 


i 


iV 


.1 


ii 


.06 


1.2 


3* 


i 


.7 


tV 


U 


1 .1166 


if 


.07 


1.4 


4 


1 


.8 


i 


1 


! .1333 


u 


.08 


1.6 


4* 


U 


.9 


tV 


u 


1 .15 


n 


.09 


1.8 


5 


u 


1.0 


« 


1 5 


.1666 


a 


.10 


2.0 


5* 


u 


1.1 


\i 


isV 


i .1833 


n 


.11 


2.2 


6 


14 


1.2 


i 


li 


1 .2 


u 


.12 


2.4 



MECHANICAL DRAWING. 



119 



PROPORTIONS 

OF 

NUTS AND B0LT5 





DIAM. 
OF 


THREADS 


/ \ 


r^ 




r : 1 


. PER 


\ / 




1 


BOLT. 


INCH. 


\ /. 


\^ 


' 


1 


20 


tV 


i 


i 


A 


18 


w 


if 


1 9 


f 


16 


If 


ii 


*i 


tV 


14 


fi 


If 


If 


i 


13 


1 


I 


tV 


j\ 


12 


i.V 


3 1 
37 


li 


i 


11 


lA 


ItV 


ii 


i 


10 


ItV 


U 


5 


i ■ 


9 


iH 


ItV 


If 




8 


11 


11 


if 


u ■ 


7 


2jV 


1+1 


II 


11 


7 


2tV 


2 


1 


If 


6 


2* 


2tV 


\h 


li 


6 


2i 


n 


lA 


If 


5i 


211 


2tV 


lA 


11 


5 


StV 


21 


IS 


IJ 


5 


3M 


2}f 


IJI 


2 


4i 


31 


34 


ItV 


2i 


4^ 


4,V 


3* 


15 


2i 


4 


41 


3* 


lU 


2i 


4 


4!;^ 


4i 


2J 


3 


3i 


53 


4-J 


2^^^ 



^i' 











MECHANICAL DRAWING. 
AREAS OF CIRCLES. Area = 3.1416 X Radius^. 



131 





Diam 


Area. 


Diam 


. : Area. 


! 

Diam 


Area. 


1 Diam 


Area. 


14 


.04908 


i 


[^ 20.629 


13 


132.73 


2 


989.8 


A 


.0767 


H 


21.647 


V2 


143.14 


• 36 


1017.9 


Vi 


.11045 


1 n 


22.690 


14 


153.94 


i 2 


1046.3 


t\ 


.15033 


' 'A 


23.758 


2 


165.13 


37 


1075.2 


Jf 


.19635 


Vs 


24.850 


15 


176.71 


1 2 


1104.5 


t\ 


.24850 


H 


25.967 


2 


188.69 


38 


II34.I 


H 


.30679 


.'s 


27.108 


16 


201.06 


2 


1164.1 


H 


.37122 


6 


i 28.274 


2 


213.82 


39 


1194.6 


H 


.44179 


Ys 


29.465 


17 


226.98 


2 


1225.4 


1 3 


.51849 


1 ^ 


i 30.68 


2 


240.53 


40 


1256.6 


/8 


.60I32 


1 y. 


I 31.92 


18 


25447 


2 


1288.2 


n 


.69029 


Y2 


33.183 


2 


268.80 


41 


1320.2 


I 


.7854 


Yi 


34.471 


19 


283.53 


2 


1352.6 


ni 


.99402 


u 


35.785 


2 


298.65 


42 


1385.4 


i,¥ 


1.2272 


% 


37.122 


20 


314.16 


2 


1418.6 


i>i 


1.4849 


7 


38.484 


2 


330.06 


^3 


1452.2 


^'A 


r.7671 


^8 


39.871 


21 


346.36 


2 


1486.1 


iH 


2.0739 


, '^ 


41.282 


2 


363.05 


44 


1520.5 


^y* 


2.4053 


' Y% 


42.718 


22 


380.13 


2 


1555.3 


1.^8 


2.7612 


Y2 


44.179 


2 


397-61 


45 


1590.4 


2 


3..1416 


Ys 


45.663 


23 


415.48 


2 


1625.9 


% 


3.5466 


¥ 


47.173 


2 


433.74 


46 


1661.9 


H 


3.9761 


'» 


48.707 


24 


452.39 


2 j 


1698.2 


Vi 


4.4301 


8 


50.265 


2 


471.43 


47 i 


1734-9 


Yi 


4.9087 


Yi 


51.849 


25 


490.87 


2 1 


1772. 


H 


5-4^19 


H 


53.456 1 


2 


510.70 


48 , 


1809.5 


H 


5.9396 


Yi 


55.08S , 


26 


530.93 


2 


1847.4 


'» 


6.4918 


Yz 


56.745 


2 


551.55 


49 


1885.7 


3 


7.0686 


Y^ 


58.426 


27 


572.56 


2 


1924.4 


% 


7.6699 


Y 1 


60.132 


2 


593. 9^^ 


50 


1963 5 


H 


8.2958 


'« 1 


61.862 


28 


615.75 


2 ; 


2002.9 


Y% 


8.9462 


9 


63-617 1 


2 


637.94 


51 ■ 


2042.8 


)o 


9.621 1 


Y^ 


65.397 1 


29 


660 52 


2 ' 


2083. 


% 


10.320 


y* . 


67.201 1 


2 


683.49 


52 ! 


2123.7 


'i 


TI.045 


H 


69.029 


30 


706.86 


2 


2164.7 


'8 


11.793 


V-. 


70.882 : 


2 


730.62 


53 : 


2206.2 


4 


12.566 


% 


72.76 


31 


754.77 


2 


2248. 


% 


13.364 


Y, 


74.66 


2 


779.31 


54 i 


2290.2 


H 


14.186 


7 
8 


76.59 


32 


804.25 , 


2 ' 


2332.8 


/s 


15-033 


10 


78.54 


2 


829.58 


55 ; 


2375-8 


'•> 


15.904 


'A 


86.590 


33 


855-30 


2 


2419.2 


% 


16.800 


II 


95.033 


2 


8S1.41 


56 


2463. 


■f 


17.72 


% 


103.87 


34 


i)()7 92 


2 


2507.2 


'a 


1S.665 


12 


113. 1 


2 


<)34.82 , 


57 


2551.7 


5 


19.635 


Vz 


122.72 


35 


962.11 1 


2 1 


2596.7 



122 



MECHANICAL DRAWING. 



AREAS OF CIRCLES.— Continued. 



Diam. 


! 

Area. 


j 
Diam. 


A.ea. 1 


1 

Diam. 


Area. 


Diam. 


Area. 


58 


2642.1 


69 


3739.3 


. 2 


4963.9 


90 


6361.7 


2 


2687.8 


2 


3793.7 


80 


5026.5 j 


2 


6432.6 


59 


2734. ' 


70 


3848.4 


2 


5089.6 ' 


91 


6503.9 


2 


2780.5 


1 2 


3903.6 


81 


5153. 


2 


6575.5 


60 


2827.4 


71 


3959.2 


2 


5216.6 


92 


6647.6 


2 


2874.7 


2 


4015. I 


82 


5281. i 


2 


6720.1 


61 


2922.5 


72 


4071.5 


2 


5345.6 


93 


6792.9 


2 


2970.6 


2 


4128.2 


83 


54^0.6 


2 


6866.1 


62 


3019.1 


, 73 


4185.4 


2 


5476. 


94 


6939.8 


2 


3067.9 


2 


4242.9 


84 


5541.78! 


! 2 


7013.8 


63 


3117.2 


74 


4300.8 


2 


5607.9 : 


95 


7088.2 


2 


3166.9 


2 


4359.1 


85 


5674.5 j 


2 


7163. 


64 


3217. 


75 


4417.8 


2 


5741.4 i 


96 


7238.2 


2 


3267.4 


2 


4477. 


86 


5808.8 ■ 


2 


7313.8 


65 


3318.3 i 


76 


4536.4 


2 


5876.5 


97 


7389.8 


2 


3369.5 1 


2 


4596.3 


87 


5944.7 


2 


7466.2 


66 


3421.2 


77 


4656.6 


2 


6013.2 


; 98 


7543. 


2 


3473.2 


2 


4717.3 


88 


6082 . I 


2 


7620.1 


67 


3525.6 


78 


4778.3 


2 


6151.4 


: 99 


7697.7 


2 


3578.5 


2 


4839.8 


89 


6221. I 1 


2 


7775.6 


68 


3631.7 1 


79 


4901.7 ! 


2 


6291.2 


100 


7854. 


2 


3685.3 1 















MECHANICAL DRAWING. 



123 



CIRCUMFERENCES OF CIRCLES. C = 3.1416 X Diameter. 



















Diam. 


! 

! Circum. 


Diam. 

% 


Circum. 


Diam. 


Circum. 
71.471 


Diam. 


1 
Circum. 


M 


.7854 


36.128 


i 

38 


I 119.38 


% 


1.5708 


y 


i 36.914 


23 


72.257 


K 


I 120.95 


H 


2.3562 


12 


: 37.699 


M 


73.042 


39 


122.52 


I 


3.1416 


W 


38.484 


\ A 


73.827 


% 


124.09 


iM 


3.927 


A 


39.270 


: y. 


74.613 


40 


125.66 


I'A 


4.7124 


H 


40.055 


1 24 


75.398 


Yi 


127.23 


1^ 


5.4978 


13 


40.841 


' y 


76.184 


41 


128.80 


2 


6.2832 


H 


41 .626 


% 


76.969 


. K 


130.37 


2,14 


7.0686 


: A 


42.412 


y 


77.754 


42 


131-95 


Vz 


7.854 , 


1 ^ 


43.19- 


25 


78.54 


% 


133.52 


% 


8.6394 1 


14 


43.982 


y 


79.325 


43 


1 135.9 


3 


9.4248 I 


H 


44.768 


^ li 


80. Ill 


'A 


136.66 


M 


10.210 


A 


45.553 


y 


80.896 


44 


138.23 


'A 


10.995 


H 


46.338 


26 


81.681 


'i 


139.8 


% 


II. 781 1 


15 


47.124 


y 


82.467 


45 


141.37 


4 


12.566 1 


W 


47.909 


Yz 


83.252 


y2 


142.94 


M 


13.352 { 


A 


48.695 


y 


84.038 


46 


144.51 


'A 


14.137 


% 


49.480 


27 


84.823 


li 


146.08 


H 


14.922 


16 


50.265 


y 


85.608 


47 


147.65 


5 


15.708 


i H 


51.051 


Vz 


86.394 


Yi 


149.22 


^4: 


16.493 


' A 


51.836 


■ y 


87.179 


48 


150.8 


'^ 


17.279 


y 


52.622 


28 


87.965 


Vi 


152.37 


y* 


18.064 


17 


53.407 


y 


88.75 


49 


153.94 


6 


18.849 


M 


54.192 


Yz 


89.535 


Vz 


155.51 


¥ 


19.635 i 


Yi 


54.978 


y 


90.321 


50 


157.08 


^ 


20.420 


H 


55.763 


29 


91 .106 


% 


I5S.65 


14' 


21.206 ! 


18 


56.549 


y 


91.892 


51 


160.22 


7 


21.991 1 


H : 


57.334 


. A 


92.677 


'A 


161.79 


,^4' 


22.776 ! 


'A 


58.119 


■ y 


93.462 


52 


163.36 


>^ 


23.562 ' 


Va 


58.905 


30 


94.248 


Vz 


164.93 


14' 


24.347 i 


19 I 


59.69 


Yz 


95.819 


53 


166.5 


8 


25.133 1 


H. i 


60.476 


31 


97.389 


V2 


168.07 


'i 


25:918 1 


li 1 


61.261 


Yz 


98.96 


54 


169.64 


A 


26.703 ! 


M 


62 . 046 


32 


100 53 


'A 


171.22 


H 


27.489 : 


20 


62.832 


^■i 


102.10 1 


55 


172.79 


9 


28.274 


H \ 


63.617 


33 


103.67 


% 


174.36 


^4 


29.060 


A , 


64.403 


A 


105.24 


56 


175-93 


3-2 


29.845 


% 1 


65.188 


34 


106.81 


J/i 


177.5 


«' 


30.630 


21 


65.973 


!•> 


108.38 


57 


179.07 


10 


31.416 ; 


\^ i 


66.759 


35 


109.95 


A \ 


180.64 


'4 


32.201 ! 


% I 


67 544 


Vz 


III. 53 


58 


182.21 


>^ 


32.987 i 


y* \ 


68.330 


36 


113. 10 


;^ 1 


183.78 


H 


33.772 ■ 


22 


69 . 1 1 5 


1., 


114.67 


59 


185.35 


11 


34.557 


% ' 


69.9 


37 


116.24 


Vi 


186.92 


''4 


35.343 1 


A 1 


70.686 


Yz 


117. Si 


60 


188.49 



124 



MECIIAXICAL DRAWING . 



CIRCUMFERENCES OF CIRCLES. Continued. 





Diam. 


Circum. 


Diam 


Circum. 


I 
|Diam. 


1 
Circum. 


Diam. 


Circum. 


'A 


190.07 


% 


221 .48 


2 


252.90 


2 


284.31 


6i 


191.64 


71 


223.05 


1 81 


254.47 


1 91 


285.88 


% 


193.21 


V2 


224.62 


' 1 


256.04 


2 


287.45 


62 


194.78 


72 


226.19 


82 


257.61 


92 


289.03 


^., 


10.35 


2 


227.76 


2 


259.18 


2 


290 . 6 


63 


197.92 


73 


229.34 


' S3 


260.75 


93 


292.17 


1., 


199.49 


2 


230.91 


2 


262.32 


2 


293 . 74 


64 


201 06 


74 


232.48 


S4 


263.89 


94 


295.31 


% 


202 . 63 


2 


234.05 


2 


265.46 j 


2 


296.88 


^^5 


204 . 20 


75 


235.62 


85 


267.03 


95 


298.45 


2 


205.77 


2 


237.19 


2 


268.61 


2 


300 . 02 


66 


207.34 


76 


238.76 


86 


270.18 


96 


301.59 


2 


208.91 





240.33 


2 


271.75 


2 


303.16 


67 


210.49 


77 


241.9 


87 


273.32 


97 


304.73 


2 


212.06 


2 


243.47 


2 


274.89 


2 


306.3 


68 


213.63 


7S 


245.04 


8*8 


276.46 


98 


307.88 


•^ 


215.2 


2 


246.61 


2 


278.03 


2 


309.45 


69 


216.77 


79 


248.18 


89 


279.60 


99 


3 1 1 . 02 


1/ 

72 


218.34 


2 


219.76 


2 


281.17 


2 


312.59 


70 


219.91 


So 


251.33 


90 


282.74 _ 


100 


314.16 



[0 019 945 477 8 



/-v. 



